<Text-field style="Heading 1" layout="Heading 1">Introdu\303\247\303\243o</Text-field>

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Cap\303\255tulo 11

Regras de Derivada

Calcular derivadas usando a defini\303\247\303\243o depende da habilidade para avaliar o limite do quociente das diferen\303\247as. Isto \303\251 tedioso e demorado. E pior, os limites podem ser extremamente dif\303\255ceis de serem avaliados. As regras de difrerencia\303\247\303\243o s\303\243o uma cole\303\247\303\243o de regras gerais para calcular derivadas formadas de polin\303\264mios, pot\303\252ncias, raizes, fun\303\247\303\265es multiplicadas por constantes, somas, diferen\303\247as, produtos, quocientes desses tipos de fun\303\247\303\265es. Composi\303\247\303\265es de fun\303\247\303\265es s\303\243o manipuladas separadamente no pr\303\263ximo cap\303\255tulo, na regra da cadeia.

Tenha cuidado para n\303\243o confundir regras de limites com regras de deriva\303\247\303\243o. Enquanto muitas s\303\243o an\303\241logas, outras s\303\243o completamente diferentes.

O maplet Differentiation \303\251 uma ferramenta excelente para aprender os nomes e fun\303\247\303\265es das regras de deriva\303\247\303\243o e ensinar como elas podem ser aplicadas para avaliar uma derivada. Os tr\303\252s exemplos apresentados nesta planilha introduzir\303\243o no uso dessa ferramenta.

<Text-field style="Heading 1" layout="Heading 1">\303\215ndice de Termos</Text-field>

Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio do Maple; para mais informa\303\247\303\265es v\303\241 em Dicion\303\241rio Matem\303\241tico.

suposi\303\247\303\243o

regra da cadeia

constante

cont\303\255nuo

derivada

quociente das diferen\303\247as

diferen\303\247a

diferenci\303\241vel

diferenciada

diferencia\303\247\303\243o

evaliar

par

expl\303\255cita

expoente

fun\303\247\303\243o

gr\303\241fico

limite

linear

ponto m\303\251dio

m\303\272ltiplo

negativo

numerador

origem

polinomial

positivo

regra da pot\303\252ncia

regra do produto

produto

prova

regra do quociente

quociente

taxa de varia\303\247\303\243o

fun\303\247\303\243o racional

restante

inclina\303\247\303\243o

vari\303\241vel

<Text-field style="Heading 1" layout="Heading 1">Inicializa\303\247\303\243o</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

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

<Text-field style="Heading 1" layout="Heading 1">As Regras de Deriva\303\247\303\243o</Text-field>

Em geral, derivadas podem ser avaliadas aplicando a defini\303\247\303\243o, como nos Exemplos 10.1 - 10.3 no cap\303\255tulo anterior. As primeiras cinco regras listadas na Tabela 11.1 s\303\243o bastante simples, quase \303\263bvias. Mas, come\303\247ando com a Regra do Produto, as regras n\303\243o s\303\243o t\303\243o \303\263bvias e s\303\243o bastante diferentes das regras de limites.

Nome da Regra

Regra de Deriva\303\247\303\243o

Condi\303\247\303\265es

Constante

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

Identidade

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

Pot\303\252ncia

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 um inteiro positivo

Multiplicada por Constante

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

Soma

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2KkYrLUYjNiYtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y9USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZWLUYwNiQtRiM2JC1GLDYlUSJ4RidGOUY8RkNGQ0ZDLUZANi1RMSZJbnZpc2libGVUaW1lcztGJ0ZDRkVGSEZKRkxGTkZQRlJGVEZXLUZANi1RJyZwbHVzO0YnRkNGRUZIRkpGTEZORlBGUkZURldGam4tRiM2Ji1GLDYlUSJnRidGOUY8Rj9GWUZDRitGQ0ZDLUZANi1RIidGJ0ZDRkVGSEZKRkxGTkZQRlJGVEZXRmpuLUZANi1RIj1GJ0ZDRkVGSEZKRkxGTkZQRlJGVEZXRmpuRjZGam5GZW9GWUZqbkZdb0ZqbkZib0Zlb0ZZRitGQw==

Diferen\303\247a

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

Produto

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

Quociente

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

Tabela 11.1

Regras de Deriva\303\247\303\243o: os operadores LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= e linha (') s\303\243o usados, tanto um quanto o outro, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= representa uma constante, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= s\303\243o fun\303\247\303\265es diferenci\303\241veis em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

Observe as condi\303\247\303\265es listadas na terceira coluna. Se as condi\303\247\303\265es para uma regra n\303\243o forem satisfeitas, a regra n\303\243o pode ser usada para avaliar a derivada.

A regra Multiplicada por Constante simplesmente diz que, ao derivar um produto de uma fun\303\247\303\243o vezes uma constante, computa-se a derivada da fun\303\247\303\243o e multiplica-se pela constante.

As regras de Soma e Diferen\303\247a simplesmente dizem que a derivada de uma soma ou diferen\303\247a \303\251 a soma ou diferen\303\247a das derivadas.

A regra do Produto pode ser melhor entendida como "o primeiro pela derivada do segundo mais o segundo pela derivada do primeiro."

Na realidade, o padr\303\243o a ser seguido \303\251 melhor entendido se a derivada de um produto de tr\303\252s fatores for escrita. Para isto temos:

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O produto de tr\303\252s fatores \303\251 escrito tr\303\252s vezes, em cada um destes tr\303\252s termos, somente uma das fun\303\247\303\265es \303\251 derivada, e os termos s\303\243o somados. Com este padr\303\243o em mente, a Regra do Produto \303\251 vista como obedecendo exatamente a este esquema, mas a vocaliza\303\247\303\243o em termos de "primeiro" e "segundo" faz a articula\303\247\303\243o e aplica\303\247\303\243o da regra muito mais simples.

Finalmente, o quociente pode ser colocado como "denominador vezes a derivada do numerador, menos numerador vezes a derivada do denominador, tudo isto sobre o denominador ao quadrado." (Com esta vers\303\243o da regra, as letras DN lideram a vocaliza\303\247\303\243o. Relembrando que D representa denominador e que DNA \303\251 a chave da vida biol\303\263gica, nenhum estudante deveria jamais trocar os pap\303\251is de numerador e denominador na regra do Quociente.)

Em algumas classes, estudantes aprendem a regra do Quociente com "de cima" e "de baixo", recolocando "numerador" e "denominador," respectivamente. Em qualquer caso, articulando um padr\303\243o em linguagem, \303\251 muito mais f\303\241cil que tentar memorizar isto como uma sequ\303\252ncia de s\303\255mbolos.

<Text-field style="Heading 1" layout="Heading 1">Provas</Text-field>

\303\211 muito dif\303\255cil dar uma prova de regras de deriva\303\247\303\243o usando Maple. Em alguns casos \303\251 poss\303\255vel dar um argumento geom\303\251trico usando um construtor de gr\303\241fico do Maple, em outros casos, o Maple pode ser usado para identificar o quociente da diferen\303\247a como derivadas. Por\303\251m, deve-se ter muito cuidado se para distinguir entre uma prova e um argumento baseado em "porque o Maple disse assim."

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova de Regra da Constante </Text-field>

A Regra da Constante pode ser entendida ao notar que o gr\303\241fico de uma fun\303\247\303\243o constante \303\251 uma reta horizontal, i.e., tem inclina\303\247\303\243o 0. (Veja Figura 11.1.)

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

LSUlUExPVEc2Ji0lJ0NVUlZFU0c2JDdkdzckJCEjSSEiIiQiI0IhIiI3JCQhMERcKXBWWG9IISM5JCIjQiEiIjckJCEwa0dkbzE1JUghIzkkIiNCISIiNyQkITBPclUqKVEsIkghIzkkIiNCISIiNyQkITBHYjVcbSF6RyEjOSQiI0IhIiI3JCQhMHVaJnA8OVtHISM5JCIjQiEiIjckJCEwTnFTZnElPkchIzkkIiNCISIiNyQkITAiR2NfS3kqeSMhIzkkIiNCISIiNyQkITBhMjpjITNmRiEjOSQiI0IhIiI3JCQhMCc9UE1qWkdGISM5JCIjQiEiIjckJCEwOERdNycqcHAjISM5JCIjQiEiIjckJCEwXChcUiRvI3BFISM5JCIjQiEiIjckJCEwXiwuVWAhUUUhIzkkIiNCISIiNyQkITBDWydILnIxRSEjOSQiI0IhIiI3JCQhME5xU1gwbGQjISM5JCIjQiEiIjckJCEwJG9PTGwyXEQhIzkkIiNCISIiNyQkITBNb08qM1k7RCEjOSQiI0IhIiI3JCQhMEFWJ282JCkpWyMhIzkkIiNCISIiNyQkITBEXStCInBjQyEjOSQiI0IhIiI3JCQhMFkjXFE+Q0dDISM5JCIjQiEiIjckJCEwVSRvYyZHcVIjISM5JCIjQiEiIjckJCEvJXplLjF0TyMhIzgkIiNCISIiNyQkITBmPE5xJEhPQiEjOSQiI0IhIiI3JCQhMC8yOXE5eUkjISM5JCIjQiEiIjckJCEwRV4tRCc0eEEhIzkkIiNCISIiNyQkITAjUXc3JSk9WEEhIzkkIiNCISIiNyQkITA7S2thN3VAIyEjOSQiI0IhIiI3JCQhMERcKVxRVCg9IyEjOSQiI0IhIiI3JCQhMF8vNEVBazojISM5JCIjQiEiIjckJCEwJHBRUEk1RUAhIzkkIiNCISIiNyQkITBDWydId3cnNCMhIzkkIiNCISIiNyQkITBfLzRtJj5rPyEjOSQiI0IhIiI3JCQhMFkiSClIR1wuIyEjOSQiI0IhIiI3JCQhMGEzPDl6TysjISM5JCIjQiEiIjckJCEwYDE4d2lgKD4hIzkkIiNCISIiNyQkITAmND55YVNXPiEjOSQiI0IhIiI3JCQhMG9PdDN4XyI+ISM5JCIjQiEiIjckJCEwbEhmX0hbKT0hIzkkIiNCISIiNyQkITB0WSRwOTFiPSEjOSQiI0IhIiI3JCQhMEtrRzIoKlEjPSEjOSQiI0IhIiI3JCQhMFF2XTAjKVF6IiEjOSQiI0IhIiI3JCQhMDZAVU8oPWo8ISM5JCIjQiEiIjckJCEwXy80JW91SzwhIzkkIiNCISIiNyQkITAvMjk7dlpxIiEjOSQiI0IhIiI3JCQhLzA1Z2tyczshIzgkIiNCISIiNyQkITBSeGF4VVNrIiEjOSQiI0IhIiI3JCQhMG9OcmtxTWgiISM5JCIjQiEiIjckJCEwKXBSPiM0VWUiISM5JCIjQiEiIjckJCEwNkFXOz08YiIhIzkkIiNCISIiNyQkITAmND4pSCRlQjohIzkkIiNCISIiNyQkITAkb090Klw7XCIhIzkkIiNCISIiNyQkITBPc1dAWURZIiEjOSQiI0IhIiI3JCQhLy8zdzdySTkhIzgkIiNCISIiNyQkIS8yOTNKQy45ISM4JCIjQiEiIjckJCEweWI2Mmc+UCIhIzkkIiNCISIiNyQkITAwND0/VTxNIiEjOSQiI0IhIiI3JCQhMExsSTJXOkoiISM5JCIjQiEiIjckJCEwJSopeVBxWCJHIiEjOSQiI0IhIiI3JCQhLyp6ZiJHYl83ISM4JCIjQiEiIjckJCEwbkxuZTE4QSIhIzkkIiNCISIiNyQkITA3QlltRTo+IiEjOSQiI0IhIiI3JCQhMFolKikpcCU9ZzYhIzkkIiNCISIiNyQkITAiR2NzRSI9OCIhIzkkIiNCISIiNyQkIS8zO3NpWCs2ISM4JCIjQiEiIjckJCEwN0JZSUEvMiIhIzkkIiNCISIiNyQkITAtLjFXaC8vIiEjOSQiI0IhIiI3JCQhMDA1Pz9uIjQ1ISM5JCIjQiEiIjckJCEwWSJIZTFXLikqISM6JCIjQiEiIjckJCEwJXplPEJEMyYqISM6JCIjQiEiIjckJCEwKEdkOSRIQj0qISM6JCIjQiEiIjckJCEwSGU7dEtzKSkpISM6JCIjQiEiIjckJCEwLjA1P1RhZSkhIzokIiNCISIiNyQkIS8oUnopcF95IykhIzkkIiNCISIiNyQkITBjN0Q1OGsqeiEjOiQiI0IhIiI3JCQhLzQ9TykpbyZwKCEjOSQiI0IhIiI3JCQhMGM3RHFgc1IoISM6JCIjQiEiIjckJCEwJz1QdTMteXEhIzokIiNCISIiNyQkITBLakV0Nmp6JyEjOiQiI0IhIiI3JCQhMGY7TCd5RnJrISM6JCIjQiEiIjckJCEwUXdfJilmejwnISM6JCIjQiEiIjckJCEwWCE0PXF0KCllISM6JCIjQiEiIjckJCEwJGU7TFVsdmIhIzokIiNCISIiNyQkITBCWCE0ITRCRSYhIzokIiNCISIiNyQkITBHYzdENnMoXCEjOiQiI0IhIiI3JCQhMFQjWyc0XVduJSEjOiQiI0IhIiI3JCQhKVspKnlWISIpJCIjQiEiIjckJCEwYTI6SShwZVMhIzokIiNCISIiNyQkITA6SGVjZjp5JCEjOiQiI0IhIiI3JCQhMC4wNSFvRGpNISM6JCIjQiEiIjckJCEwclQkb2Eua0ohIzokIiNCISIiNyQkITA+UHVHL3gnRyEjOiQiI0IhIiI3JCQhMEtrR1BQNmQjISM6JCIjQiEiIjckJCEwSmlDKnlsb0EhIzokIiNCISIiNyQkITB4XzByTSFbPiEjOiQiI0IhIiI3JCQhMEJZI1xZMGE7ISM6JCIjQiEiIjckJCEvJioqKXomZVFPIiEjOSQiI0IhIiI3JCQhMCYqKXlkUGtgNSEjOiQiI0IhIiI3JCQhMGM2QllVNVcoISM7JCIjQiEiIjckJCEwPFF3X1RGbSUhIzskIiNCISIiNyQkITAoSGY9UGtyOCEjOyQiI0IhIiI3JCQiMHhkOkolPiNRIiEjOyQiI0IhIiI3JCQiMDhJZz88SmclISM7JCIjQiEiIjckJCIwLS0vM3dybighIzskIiNCISIiNyQkIjBCWSNcVz5VNSEjOiQiI0IhIiI3JCQiLyI+UU9zM04iISM5JCIjQiEiIjckJCIvKnpmUidmaDshIzkkIiNCISIiNyQkIjBHYjVoVjMoPiEjOiQiI0IhIiI3JCQiMDlIZU9idkQjISM6JCIjQiEiIjckJCIwXy80eUdXYiMhIzokIiNCISIiNyQkIjBHZDlwYjknRyEjOiQiI0IhIiI3JCQiMDI5Ryd6XG5KISM6JCIjQiEiIjckJCIwVCJHYytJI1skISM6JCIjQiEiIjckJCIwemQ6InpkZlAhIzokIiNCISIiNyQkIjBmPE41RjwyJSEjOiQiI0IhIiI3JCQiMERdKyxlXlElISM6JCIjQiEiIjckJCIwOUhldzFzbyUhIzokIiNCISIiNyQkIjBLa0coZlxoXCEjOiQiI0IhIiI3JCQiMERcKXBCbChHJiEjOiQiI0IhIiI3JCQiLzA1PydcUmMmISM5JCIjQiEiIjckJCIwOklnK1xgKWUhIzokIiNCISIiNyQkIjAvMzsjPiUpcGghIzokIiNCISIiNyQkIjBcKXBSZCg+WychIzokIiNCISIiNyQkIjBwUXglND96biEjOiQiI0IhIiI3JCQiMHljOEZDJCozKCEjOiQiI0IhIiI3JCQiMEppQ0g5VFAoISM6JCIjQiEiIjckJCIvLC0vKSlIIm8oISM5JCIjQiEiIjckJCIwWiUqKXlyUCshKSEjOiQiI0IhIiI3JCQiMDE2QSVlOHkjKSEjOiQiI0IhIiI3JCQiLy0vM0c3eSYpISM5JCIjQiEiIjckJCIwVihbKHBRISkpKSkhIzokIiNCISIiNyQkIjBLakUkNEIiPiohIzokIiNCISIiNyQkIjBEXSssJmUlWyohIzokIiNCISIiNyQkIjBXKFsocC8uIikqISM6JCIjQiEiIjckJCIwIj1PS3lINTUhIzkkIiNCISIiNyQkIjBzVyopKXBhVDUhIzkkIiNCISIiNyQkIjB0WSRwTCcpcDUhIzkkIiNCISIiNyQkIjBKaUNsPzM1IiEjOSQiI0IhIiI3JCQiMGU7TC9cKkg2ISM5JCIjQiEiIjckJCIwaUJaZydSZzYhIzkkIiNCISIiNyQkIjBgMThtayw+IiEjOSQiI0IhIiI3JCQiMCYqKXlkIUg4QSIhIzkkIiNCISIiNyQkIjAqeWR2U01eNyEjOSQiI0IhIiI3JCQiMDtLa3dRP0ciISM5JCIjQiEiIjckJCIwdVsoKkd6Q0oiISM5JCIjQiEiIjckJCIwQlkjcDRYUzghIzkkIiNCISIiNyQkIjB3XzBuNERQIiEjOSQiI0IhIiI3JCQiMCllPGJMPSw5ISM5JCIjQiEiIjckJCIwZjxOW2I8ViIhIzkkIiNCISIiNyQkIjBHYzcicCxoOSEjOSQiI0IhIiI3JCQiMDtKaSd6XSRcIiEjOSQiI0IhIiI3JCQiMEppQyRHa0A6ISM5JCIjQiEiIjckJCIwVidHZGhkYDohIzkkIiNCISIiNyQkIi80PTsqekVlIiEjOCQiI0IhIiI3JCQiMCdHZGFbXjk7ISM5JCIjQiEiIjckJCIwYzdELSQpPmsiISM5JCIjQiEiIjckJCIwXChcZmdFdDshIzkkIiNCISIiNyQkIjBBVylHUlsuPCEjOSQiI0IhIiI3JCQiMCV6ZWQ/b0w8ISM5JCIjQiEiIjckJCIwS2tHNHBQdyIhIzkkIiNCISIiNyQkIjBQdFlKdEV6IiEjOSQiI0IhIiI3JCQiLyc+UmE+UiM9ISM4JCIjQiEiIjckJCIwOklnWSpwYD0hIzkkIiNCISIiNyQkIjB6ZTxWVF0pPSEjOSQiI0IhIiI3JCQiMFghNGVNVDg+ISM5JCIjQiEiIjckJCIwWSNcZSlwWiU+ISM5JCIjQiEiIjckJCIwOklnI1EhWyg+ISM5JCIjQiEiIjckJCIwRF0rcGtaKyMhIzkkIiNCISIiNyQkIjBBVidHKmVnLiMhIzkkIiNCISIiNyQkIjA3Q1sxIylbMSMhIzkkIiNCISIiNyQkIjBaJSopKSozUyU0IyEjOSQiI0IhIiI3JCQiMClmPio+JCpwNyMhIzkkIiNCISIiNyQkIjBXKFtkR11jQCEjOSQiI0IhIiI3JCQiMHdfMCwjbyc9IyEjOSQiI0IhIiI3JCQiLyRmPVZ0dEAjISM4JCIjQiEiIjckJCIwLC0vI1tlWEEhIzkkIiNCISIiNyQkIjA9TnFDZGNGIyEjOSQiI0IhIiI3JCQiMCwtL3crYkkjISM5JCIjQiEiIjckJCIwMztLL0N1TCMhIzkkIiNCISIiNyQkIjAkcFFkXGZsQiEjOSQiI0IhIiI3JCQiMGhAVk0pNClSIyEjOSQiI0IhIiI3JCQiMGpEXjlJdVUjISM5JCIjQiEiIjckJCIwQVcpR0NYY0MhIzkkIiNCISIiNyQkIjBvT3RxZ3dbIyEjOSQiI0IhIiI3JCQiMHVbKEhfKio9RCEjOSQiI0IhIiI3JCQiMGtGYiswdmEjISM5JCIjQiEiIjckJCIwLjA1NyJ5eEQhIzkkIiNCISIiNyQkIjBGYDFsRnRnIyEjOSQiI0IhIiI3JCQiMF4tMFNjJFJFISM5JCIjQiEiIjckJCIwTnFTPHFxbSMhIzkkIiNCISIiNyQkIjB3XzBYKyopcCMhIzkkIiNCISIiNyQkIi8iPlFlQSlHRiEjOCQiI0IhIiI3JCQiMGI0PnFiJWVGISM5JCIjQiEiIjckJCIwJG9PJFI3Iil5IyEjOSQiI0IhIiI3JCQiMC4yOU1nJD1HISM5JCIjQiEiIjckJCIwKnpmZkVVXUchIzkkIiNCISIiNyQkIjBrR2RtPyl6RyEjOSQiI0IhIiI3JCQiMEtqRUZTKTNIISM5JCIjQiEiIjckJCIwUHVbdmgpUkghIzkkIiNCISIiNyQkIjA6SWdxOjMoSCEjOSQiI0IhIiI3JCQiI0khIiIkIiNCISIiLSUmQ09MT1JHNiclJFJHQkckIiM1ISIiJCIiISEiIiQiIiEhIiItJStfQVRUUklCVVRFRzYjLyUnc291cmNlRyUsbWF0aGRlZmF1bHRHLSUlVklFV0c2JDskISNJISIiJCIjSSEiIiUoREVGQVVMVEctJStBWEVTTEFCRUxTRzYkLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEieDYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiUSE2Ii0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJEkiISIiLSUpQk9VTkRTX1lHNiMkIiNdISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlIXokISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJT9OISIiLSUpQ0hJTERSRU5HNiI=

Figura 11.1 Uma fun\303\247\303\243o constante tem inclina\303\247\303\243o 0

A defini\303\247\303\243o de uma fun\303\247\303\243o constante \303\251 simples de aplicar. Definir a fun\303\247\303\243o constante 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 . E, ent\303\243o, aplicar a defini\303\247\303\243o de derivada e simplificar, obtendo:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiMEYnRjVGNUYrRjU=

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova da Regra da Identidade</Text-field>

A Regra da Identidade pode ser tirada do nosso conhecimento de fun\303\247\303\265es lineares. O gr\303\241fico da fun\303\247\303\243o 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 \303\251 uma linha reta que passa pela origem com inclina\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Assim, 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.

Uma prova rigorosa baseada na defini\303\247\303\243o tamb\303\251m n\303\243o \303\251 dif\303\255cil. Sendo a fun\303\247\303\243o identidade LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0Y2RjlGQEZARkAtRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yNzc3Nzc4ZW1GJy9GVUZcb0ZlbkZARitGQA==,

temos:

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova da Regra de Pot\303\252ncia (n inteiro positivo)</Text-field>

Primeiro, note que a Regra da Pot\303\252ncia com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= \303\251 a Regra da Identidade, a saber,

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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIwRicvRjtRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdGQEYrRkFGK0ZB = 1

Vimos tamb\303\251m que (Exemplo 10.1, Cap\303\255tulo 10) que 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, que \303\251 a Regra da Pot\303\252ncia \303\251 mantida paraLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RIj1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjNGJ0Y1RjVGK0Y1 .

A prova da Regra de Pot\303\252ncia quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 um inteiro positivo \303\251 baseada no padr\303\243o observado em:

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

LywmSSJ4RzYiIiIiSSJoR0YlRiZGIw==

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIiNGKCwoKiQpRiZGKkYoRigqKEYqRihGJkYoRilGKEYoKiQpRilGKkYoRig=

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIiRGKCwqKiQpRiZGKkYoRigqKEYqRigpRiYiIiNGKEYpRihGKCooRipGKEYmRigpRilGMEYoRigqJClGKUYqRihGKA==

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIiVGKCwsKiQpRiZGKkYoRigqKEYqRigpRiYiIiRGKEYpRihGKCooIiInRigpRiYiIiNGKClGKUY0RihGKCooRipGKEYmRigpRilGMEYoRigqJClGKUYqRihGKA==

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIiZGKCwuKiQpRiZGKkYoRigqKEYqRigpRiYiIiVGKEYpRihGKCooIiM1RigpRiYiIiRGKClGKSIiI0YoRigqKEYyRigpRiZGNkYoKUYpRjRGKEYoKihGKkYoRiZGKClGKUYwRihGKCokKUYpRipGKEYo

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIidGKCwwKiQpRiZGKkYoRigqKEYqRigpRiYiIiZGKEYpRihGKCooIiM6RigpRiYiIiVGKClGKSIiI0YoRigqKCIjP0YoKUYmIiIkRigpRilGOkYoRigqKEYyRigpRiZGNkYoKUYpRjRGKEYoKihGKkYoRiZGKClGKUYwRihGKCokKUYpRipGKEYo

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIihGKCwyKiQpRiZGKkYoRigqKEYqRigpRiYiIidGKEYpRihGKCooIiNARigpRiYiIiZGKClGKSIiI0YoRigqKCIjTkYoKUYmIiIlRigpRikiIiRGKEYoKihGOEYoKUYmRjxGKClGKUY6RihGKCooRjJGKClGJkY2RigpRilGNEYoRigqKEYqRihGJkYoKUYpRjBGKEYoKiQpRilGKkYoRig=

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIilGKCw0KiQpRiZGKkYoRigqKEYqRigpRiYiIihGKEYpRihGKCooIiNHRigpRiYiIidGKClGKSIiI0YoRigqKCIjY0YoKUYmIiImRigpRikiIiRGKEYoKigiI3FGKClGJiIiJUYoKUYpRkBGKEYoKihGOEYoKUYmRjxGKClGKUY6RihGKCooRjJGKClGJkY2RigpRilGNEYoRigqKEYqRihGJkYoKUYpRjBGKEYoKiQpRilGKkYoRig=

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIipGKCw2KiQpRiZGKkYoRigqKEYqRigpRiYiIilGKEYpRihGKCooIiNPRigpRiYiIihGKClGKSIiI0YoRigqKCIjJSlGKClGJiIiJ0YoKUYpIiIkRihGKCooIiRFIkYoKUYmIiImRigpRikiIiVGKEYoKihGPkYoKUYmRkJGKClGKUZARihGKCooRjhGKClGJkY8RigpRilGOkYoRigqKEYyRigpRiZGNkYoKUYpRjRGKEYoKihGKkYoRiZGKClGKUYwRihGKCokKUYpRipGKEYo

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIzVGKCw4KiQpRiZGKkYoRigqKEYqRigpRiYiIipGKEYpRihGKCooIiNYRigpRiYiIilGKClGKSIiI0YoRigqKCIkPyJGKClGJiIiKEYoKUYpIiIkRihGKCooIiQ1I0YoKUYmIiInRigpRikiIiVGKEYoKigiJF8jRigpRiYiIiZGKClGKUZGRihGKCooRj5GKClGJkZCRigpRilGQEYoRigqKEY4RigpRiZGPEYoKUYpRjpGKEYoKihGMkYoKUYmRjZGKClGKUY0RihGKCooRipGKEYmRigpRilGMEYoRigqJClGKUYqRihGKA==

Em particular, note que 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 . (Dizemos que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNictRiw2JVEiT0YnL0Y1RkJGPi1GOzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGaG4tRiM2Ji1GOzYtUSIoRidGPi9GQUY2RkMvRkZGNkZHRklGS0ZNL0ZQUSwwLjE2NjY2NjdlbUYnL0ZTRmJvLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSJoRidGNEY3LUYsNiVRInBGJ0Y0RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPi1GOzYtUSIpRidGPkZfb0ZDRmBvRkdGSUZLRk1GYW8vRlNRLDAuMTExMTExMWVtRidGPkYrRj5GK0Y+RitGPg== quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KkYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1GLDYlUSJoRicvJSdpdGFsaWNHRkgvRjlRJ2l0YWxpY0YnLUY1Ni1RJyZyYXJyO0YnRjhGO0Y+RkBGQkZEL0ZHRj1GSUZLL0ZPRk0tSSNtbkdGJDYkUSIwRidGOEY4LyUsYWNjZW50dW5kZXJHRj1GKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC40ZW1GJy8lJmRlcHRoR0Zkby8lKmxpbmVicmVha0dRJWF1dG9GJ0YrLUYjNiYtRjU2LVEiKEYnRjgvRjxGSEY+L0ZBRkhGQkZERmduRkkvRkxGUEZOLUkmbWZyYWNHRiQ2KC1GIzYkLUYsNiVRImdGJ0ZWRlhGOC1GIzYmRistSSVtc3VwR0YkNiVGUy1GLDYlUSJwRidGVkZYLyUxc3VwZXJzY3JpcHRzaGlmdEdGXG9GK0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zcci8lKWJldmVsbGVkR0Y9LUY1Ni1RIilGJ0Y4RmJwRj5GY3BGQkZERmduRklGZHAvRk9RLDAuMTExMTExMWVtRidGOEYrRjhGK0Y4 existe; em particular, 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 significa LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNihGKy1JJ211bmRlckdGJDYlLUkjbW9HRiQ2LVEkbGltRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtRiw2JVEiaEYnLyUnaXRhbGljR0ZKL0Y7USdpdGFsaWNGJy1GNzYtUScmcmFycjtGJ0Y6Rj1GQEZCRkRGRi9GSUY/RktGTS9GUUZPLUkjbW5HRiQ2JFEiMEYnRjpGOi8lLGFjY2VudHVuZGVyR0Y/RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNGVtRicvJSZkZXB0aEdGZm8vJSpsaW5lYnJlYWtHUSVhdXRvRictRiw2JVEiZ0YnRlhGWkY6LUY3Ni1RIj1GJ0Y6Rj1GQEZCRkRGRkZpbkZLL0ZOUSwwLjI3Nzc3NzhlbUYnL0ZRRmZwRltvRjpGK0Y6 .) A partir disso, n\303\243o \303\251 dif\303\255cil obter:

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

A Regra de Pot\303\252ncia n\303\243o \303\251 v\303\241lida quando o expoente \303\251 zero, um inteiro negativo , um n\303\272mero racional, ou um n\303\272mero irracional. O caso para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= \303\251 a Regra da Constante. As provas dos casos restantes dependem dos resultados que vir\303\243o mais tarde no curso. A primeira delas, inteiros negativos, \303\251 o Exemplo 11.4.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova da Regra Multiplicada por Constante </Text-field>

A Regra Multiplicada por Constante deve ser explorada intuitivamente antes de ser provada. Assim, sendo 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 e 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 a abscissa e ordenada, respectivamente, para f. Se 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 , ent\303\243o a ordenada para F ser\303\241:

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 = 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYjNiYtRjs2LVEiKEYnRj4vRkFGNkZDL0ZGRjZGR0ZJRktGTS9GUFEsMC4xNjY2NjY3ZW1GJy9GU0Zmbi1GIzYoRistRiM2Jy1GLDYlUSJmRidGNEY3LUY7Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj5GQEZDRkVGR0ZJRktGTUZPRlItRiM2JkZWLUYjNiZGKy1GIzYmLUYsNiVRInhGJ0Y0RjctRjs2LVEiK0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJy9GU0ZfcC1GLDYlUSJoRidGNEY3Rj5GK0Y+LUY7Ni1RIilGJ0Y+RllGQ0ZaRkdGSUZLRk1GZW4vRlNRLDAuMTExMTExMWVtRidGPkYrRj4tRjs2LVEoJm1pbnVzO0YnRj5GQEZDRkVGR0ZJRktGTUZecEZgcC1GIzYnRlxvRl9vLUYjNiZGVi1GIzYkRmhvRj5GZHBGPkYrRj5GK0Y+RmRwRj5GK0Y+RitGPg== = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJyYjOTE2O0YnL0Y1RkJGPkY6LUYsNiVRInlGJ0Y0RjdGPkYrRj4=

Ent\303\243o, 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 para todo 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 > 0 e assim:

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 = 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 = 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

Uma prova rigorosa baseada na defini\303\247\303\243o tamb\303\251m n\303\243o \303\251 dif\303\255cil, j\303\241 que para 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, temos:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova da Regra da Soma </Text-field>

A Regra da Soma faz sentido intuitivamente. A taxa de varia\303\247\303\243o da soma de duas fun\303\247\303\265es deve ser a soma da taxa de varia\303\247\303\243o delas (de cada uma delas), como em:

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

A prova formal da Regra da Soma segue da defini\303\247\303\243o de derivada e da Regra da Soma para limites, como vemos de:

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

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova da Regra da Diferen\303\247a </Text-field>

Uma prova da Regra da Diferen\303\247a pode ser dada exatamente da mesma forma como para a Regra da Soma. Uma alternativa \303\251 usar ambas as Regras de Multiplicada por Constante e da Soma.

Para provar a Regra da Diferen\303\247a deste modo alternativo, sejam f e g sejam fun\303\247\303\265es diferenci\303\241veis arbitr\303\241rias e escreva:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = 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 '

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<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova da Regra do Produto </Text-field>

\303\201 primeira vista, poderia se pensar na Regra do Produto como sendo simplesmente que a derivada de um produto \303\251 o produto das derivadas. Isto est\303\241 errado! Um simples exemplo para ilustrar este equ\303\255voco \303\251 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 com 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 = 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. Ent\303\243o 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 = 1, mas:

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 = 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 = 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

A prova da Regra do Produto requer um pouco de trabalho extra para colocar o quociente da diferen\303\247a para o produto, de modo que as derivadas de f e g possam ser identificadas. A chave \303\251 adicionar e subtrair a mesma quantidade de numerador, tal que quocientes das diferen\303\247as para f e g possam ser isolados em pares de termos separados. Este \303\251 um esquema padr\303\243o em an\303\241lise, e nos serve bem aqui. A defini\303\247\303\243o da derivada do produto leva a:

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

Note que 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 porque a suposi\303\247\303\243o de que f \303\251 diferenci\303\241vel em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, implica que f \303\251 cont\303\255nua em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova da Regra do Quociente </Text-field>

A prova da Regra do Quociente \303\251 similar \303\240 prova da Regra do Produto no que envolve a adi\303\247\303\243o e a subtra\303\247\303\243o da mesma quantidade do quociente da diferen\303\247a. Esta quantidade \303\251 escolhida de modo que os quocientes das diferen\303\247as de f e g possam ser identificados. Aplica\303\247\303\243o da defini\303\247\303\243o da derivada do quociente leva a:

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<Text-field style="Heading 1" layout="Heading 1">Tr\303\252s Exemplos</Text-field>

Os tr\303\252s exemplos seguintes demonstram o uso destas regras de derivada no c\303\241lculo de derivadas. Dos tr\303\252s, somente o primeiro exemplo \303\251 ainda um pouco mais simples para se tentar calcular a derivada por meio da defini\303\247\303\243o.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 11.1: Use as Regras de Deriva\303\247\303\243o para avaliar<Hyperlink linktarget="Dictionary:Definition,evaluate" hyperlink="true"> </Hyperlink><Equation executable="false" style="Heading 2" input-equation="" display="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">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</Equation></Text-field>

Tabela 11.2 resume os passos que devem ser usados para avaliar a derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjNGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRj8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkovJSlzdHJldGNoeUdGSi8lKnN5bW1ldHJpY0dGSi8lKGxhcmdlb3BHRkovJS5tb3ZhYmxlbGltaXRzR0ZKLyUnYWNjZW50R0ZKLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWUY7LUZFNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y/RkhGS0ZNRk9GUUZTRlVGV0ZaLUYwNiVGMi1GPDYkUSIyRidGP0ZBLUZFNi1RJyZwbHVzO0YnRj9GSEZLRk1GT0ZRRlNGVUZXRlpGMkZeby1GPDYkUSMzLkYnRj9GPw==

Manipula\303\247\303\243o Alg\303\251brica

Remarks

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

derivada original

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

Regras de Soma e Diferen\303\247a

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

Regra da Constante aplicada

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy1JKG1mZW5jZWRHRiQ2JC1GIzYmRistSSVtc3VwR0YkNiUtRiw2JVEieEYnRjVGOC1JI21uR0YkNiRRIjNGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGWEZYLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRlgvJSZmZW5jZUdGSC8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZbcEZURi9GKy1GSjYkLUYjNiYtRmhuNi1RMSZJbnZpc2libGVUaW1lcztGJ0ZYRltvRl1vRl9vRmFvRmNvRmVvRmdvRmlvRlxwLUZPNiVGUS1GVTYkUSIyRidGWEZaRitGWEZYLUZobjYtUScmcGx1cztGJ0ZYRltvRl1vRl9vRmFvRmNvRmVvRmdvRmlvRlxwRi9GKy1GSjYkLUYjNiRGUUZYRlhGK0ZY

Regra Multiplicada por Constante aplicada

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

Regras da Identidade e de Pot\303\252ncia aplicadas

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

Aritm\303\251tica \342\207\222 Resposta final

Tabela 11.2 Deriva\303\247\303\243o de 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

Assim, obtemos o resultado:

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

como pode ser verificado usando os comandos diff (ou Diff) do Maple, implementado do teclado ou de um menu de contexto sens\303\255vel. Um enunciado matem\303\241tico completo \303\251 formado no Maple via:

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

Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwqKiQpSSJ4R0YoIiIkIiIiRi8qJkYuRi8pRi0iIiNGLyEiIkYtRi9GLkYvRi0sKEYwRi8qJiIiJ0YvRi1GL0YzRi9GLw==

\303\211 aconselh\303\241vel repetir o exemplo no maplet Differentiation, equivalente a construir no Tutor M\303\251todos de Diferencia\303\247\303\243o dispon\303\255vel no menu Ferramentas/Tutoriais/C\303\241lculo-Uma Vari\303\241vel. A op\303\247\303\243o M\303\251todos de Diferencia\303\247\303\243o nesse menu pode ser iniciado para a fun\303\247\303\243o do Exemplo 11.1 pela sintaxe

DiffTutor(q1);

Quando iniciado, aparecer\303\241 ent\303\243o como o maplet mostrado na Figura 11.2.

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Figura 11.2 Tutor M\303\251todos de Diferencia\303\247\303\243o aplicado a 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

Pressione o bot\303\243o Come\303\247ar, ent\303\243o aplique as regras de deriva\303\247\303\243o, clicando no maplet o bot\303\243o correspondente. As barras de menu fornecem um resumo de cada regra conhecida (Defini\303\247\303\243o da Regra), Ajuda, e outro modo de aplicar regras (Aplicar Regras). Note que a regra selecionada \303\251 geralmente aplicada \303\240 primeira ocorr\303\252ncia poss\303\255vel na express\303\243o; pode ser necess\303\241rio aplicar a regra v\303\241rias vezes, sucessivamente. Regras que est\303\243o completamente entendidas podem ser marcadas como "entendidas", assim, suas aplica\303\247\303\265es se tornam autom\303\241ticas.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 11.2: Use as Regras de Deriva\303\247\303\243o para avaliar<Equation executable="false" style="2D Math_12" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KUYrLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbW9HRiQ2LlEwJkRpZmZlcmVudGlhbEQ7RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGPC8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZQL0Y+USVib2xkRicvJStmb250d2VpZ2h0R0ZULUYjNidGKy1GIzYmRjYtRiw2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y+USxib2xkLWl0YWxpY0YnRlVGU0ZVRitGU0ZVLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Ziby8lKWJldmVsbGVkR0Y8LUY3Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y9RkBGQkZERkZGSEZKRkxGTkZRLUYjNictRjc2LlEiKEYnRjpGPS9GQUZqbkZCL0ZFRmpuRkZGSEZKRkwvRk9RLDAuMTY2NjY2N2VtRicvRlJGYnAtRjI2KC1GIzYnRistRiM2J0Zlbi1GNzYuUSgmbWludXM7RidGOkY9RkBGQkZERkZGSEZKRkwvRk9RLDAuMjIyMjIyMmVtRicvRlJGXnEtSSNtbkdGJDYlRl9vRlNGVUZTRlVGK0ZTRlUtRiM2J0YrLUYjNihGKy1GIzYnRistSSVtc3VwR0YkNiVGZW4tRmFxNiVRIjJGJ0ZTRlUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGU0ZVLUY3Ni5RIitGJ0Y6Rj1GQEZCRkRGRkZIRkpGTEZdcUZfcUZcckZTRlVGK0ZTRlVGXW9GYG9GY29GZW8tRjc2LlEiKUYnRjpGPUZfcEZCRmBwRkZGSEZKRkxGYXAvRlJRLDAuMTExMTExMWVtRidGU0ZVRitGU0ZVRitGU0ZV</Equation> </Text-field>

Esta derivada ser\303\241 calculada primeiro aplicando a Regra do Quociente. As derivadas restantes requerem somente as Regras da Soma, Diferen\303\247a, Constante, Identidade e Pot\303\252ncia. Portanto, come\303\247ando com a Regra do Quociente e aplicando as outras regras conforme a necessidade, temos:

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

Esta derivada concorda com a encontrada pelo Maple via:

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

Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComLCZJInRHRigiIiJGLSEiIkYtLCYqJClGLCIiI0YtRi1GMkYtRi5GLCwmKiRGL0YuRi0qKkYyRi1GK0YtKUYvRjJGLkYsRi1GLg==

L0khRzYiLCQqJiwoKiQpSSJ0R0YkIiIjIiIiRixGKyEiIiomRitGLEYqRixGLUYsKSwmRihGLEYrRixGK0YtRi0=

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 11.3: Use as Regras de Deriva\303\247\303\243o para avaliar<Hyperlink linktarget="Dictionary:Definition,evaluate" hyperlink="true"> </Hyperlink> <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW9HRiQ2LlEwJkRpZmZlcmVudGlhbEQ7RicvJSVzaXplR1EjMTZGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUUY9LUYjNiZGKy1GIzYlRjYtRiw2KFEieEYnRjovJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0ZCL0Y+USVib2xkRicvJStmb250d2VpZ2h0R0Zbb0Y9RitGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGY28vJSliZXZlbGxlZEdGQi1GNzYtUTEmSW52aXNpYmxlVGltZXM7RidGPUZARkNGRUZHRklGS0ZNRk9GUi1GIzYmRistRiM2JkYrLUklbXN1cEdGJDYlLUYjNiYtRjc2LlEiKEYnRjpGPS9GQUZnbkZDL0ZGRmduRkdGSUZLRk0vRlBRLDAuMTY2NjY2N2VtRicvRlNGanAtRiM2KUYrLUYjNiZGKy1GYHA2JUZYLUkjbW5HRiQ2J1EiNEYnRjpGZW5Gam5GXG8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPS1GNzYuUSIrRidGOkY9RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGXXItRiM2Jy1GY3E2J1EiM0YnRjpGZW5Gam5GXG9GaG8tRiM2JkYrLUZgcDYlRlgtRmNxNidRIjJGJ0Y6RmVuRmpuRlxvRmZxRitGPUYrRj1GaXEtRmNxNidGYG9GOkZlbkZqbkZcb0Y9LUY3Ni5RIilGJ0Y6Rj1GZ3BGQ0ZocEZHRklGS0ZNRmlwL0ZTUSwwLjExMTExMTFlbUYnRj1GaHJGZnFGK0Y9RitGPUYrRj1GK0Y9">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW9HRiQ2LlEwJkRpZmZlcmVudGlhbEQ7RicvJSVzaXplR1EjMTZGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUUY9LUYjNiZGKy1GIzYlRjYtRiw2KFEieEYnRjovJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0ZCL0Y+USVib2xkRicvJStmb250d2VpZ2h0R0Zbb0Y9RitGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGY28vJSliZXZlbGxlZEdGQi1GNzYtUTEmSW52aXNpYmxlVGltZXM7RidGPUZARkNGRUZHRklGS0ZNRk9GUi1GIzYmRistRiM2JkYrLUklbXN1cEdGJDYlLUYjNiYtRjc2LlEiKEYnRjpGPS9GQUZnbkZDL0ZGRmduRkdGSUZLRk0vRlBRLDAuMTY2NjY2N2VtRicvRlNGanAtRiM2KUYrLUYjNiZGKy1GYHA2JUZYLUkjbW5HRiQ2J1EiNEYnRjpGZW5Gam5GXG8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPS1GNzYuUSIrRidGOkY9RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGXXItRiM2Jy1GY3E2J1EiM0YnRjpGZW5Gam5GXG9GaG8tRiM2JkYrLUZgcDYlRlgtRmNxNidRIjJGJ0Y6RmVuRmpuRlxvRmZxRitGPUYrRj1GaXEtRmNxNidGYG9GOkZlbkZqbkZcb0Y9LUY3Ni5RIilGJ0Y6Rj1GZ3BGQ0ZocEZHRklGS0ZNRmlwL0ZTUSwwLjExMTExMTFlbUYnRj1GaHJGZnFGK0Y9RitGPUYrRj1GK0Y9</Equation></Text-field>

\303\200 primeira vista n\303\243o temos (ainda) as t\303\251cnicas necess\303\241rias para avaliar esta derivada. Mas, j\303\241 que:

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

a Regra do Produto pode ser usada. (Note que o Tutor M\303\251todos de Diferencia\303\247\303\243o n\303\243o acredita que a regra do Produto possa ser aplicada aqui. A dica sugere que a Regra da Cadeia deve ser aplicada. Como n\303\243o conhecemos ainda a Regra da Cadeia, vamos seguir este exemplo sem o uso do maplet.) Aplica\303\247\303\243o da regra do produto deste modo leva a:

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

Note que uma abordagem alternativa para este problema come\303\247a com a expans\303\243o:

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

Ent\303\243o a deriva\303\247\303\243o termo a termo leva a:

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

Para checar estes dois resultados, recorremos ao Maple para ajudar com a \303\241lgebra. Primeiro, o resultado usando a Regra do Produto pode ser resumida como:

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

Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCokKSwoKiQpSSJ4R0YoIiIlIiIiRjEqJiIiJEYxKUYvIiIjRjFGMUYxRjFGNUYxRi8sKiomIiIpRjEpRi8iIihGMUYxKiYiI09GMSlGLyIiJkYxRjEqJiIjV0YxKUYvRjNGMUYxKiYiIzdGMUYvRjFGMQ==

A deriva\303\247\303\243o obtida quando a fun\303\247\303\243o \303\251 expandida antes de ser diferenciada termo a termo \303\251:

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

Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCokKSwoKiQpSSJ4R0YoIiIlIiIiRjEqJiIiJEYxKUYvIiIjRjFGMUYxRjFGNUYxRi8tRiQ2JCwsKiQpRi8iIilGMUYxKiYiIidGMSlGL0Y9RjFGMSomIiM2RjFGLkYxRjEqJkY9RjFGNEYxRjFGMUYxRi8=

L0khRzYiLCoqJiIiKSIiIilJInhHRiQiIihGKEYoKiYiI09GKClGKiIiJkYoRigqJiIjV0YoKUYqIiIkRihGKComIiM3RihGKkYoRig=

Isto confirma nossas computa\303\247\303\265es iniciais e verifica que as duas abordagens fornecem o mesmo resultado.

Note que embora expans\303\243o e deriva\303\247\303\243o termo a termo n\303\243o foi mais dif\303\255cil que a Regra do Produto neste problema, a Regra do Produto \303\251 geralmente mais f\303\241cil de aplicar na maioria dos problemas envolvendo a derivada de uma quantidade elevada a uma pot\303\252ncia. Um m\303\251todo ainda mais eficiente para computar a derivada de uma composi\303\247\303\243o de duas fun\303\247\303\265es estar\303\241 no cap\303\255tulo Regra da Cadeia.

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

Using the Maple-graded question type, you create a question that uses Maple functionality when creating the correct answer and determining if the student's response is correct. The Maple-graded question type has an option to plot the student's response. To include the plot option, insert the corresponding task template. For information on creating questions, making assignments, and exporting Maple T.A. course modules, see Overview of Maple T.A. Integration.

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

Algorithmic variables can be used to create a question that contains randomly generated parameters. You can refer to algorithmic variables in the Question, Answer, Hints, and Feedback input regions, as well as in other variable definitions. To reference an algorithmic variable, include "`$variable`" as text or 2-D Math, for example, x := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique.

with(RandomTools): randomize(): n1 := Generate(integer(range=2..3)): r1 := Generate(choose([-10..-1,1..10])): fn2 := sort(Generate(polynom(integer(range=r1),x,degree=n1))): n2 := Generate(integer(range=2..3)): fn3 := sort(Generate(polynom(integer(range=r1),x,degree=n2))): fn4 := fn2*fn3; ans1 := simplify(diff(fn4,x));

KiYsKCokSSJ4RzYiIiIjIiIlRiUiIioiIzUiIiJGKywqKiRGJSIiJEYuRiQiIidGJUYvRipGK0Yr

LCwqJEkieEc2IiIiJSIjZyokRiQiIiQiJC8jKiRGJCIiIyIkQyRGJCIkMyQiJF0iIiIi

JSFH

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 11.1</Text-field>

The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question.

Usando a regra apropriada de derivadas, encontrar a derivada de :

f(x) = `$fn4`

JSFH

<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed.

JSFHans1;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$RESPONSE` is automatically initialized to the student's response. Using `$ANSWER` (the correct answer) and `$RESPONSE` (the student response), one example of evaluating to a Boolean value is, evalb(`$ANSWER`-`$RESPONSE`=0); Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region.

evalb(`$ANSWER`-`$RESPONSE`=0);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints.

Use the Product Rule.

Replace this text with hint 2.

Replace this text with hint 3.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea1`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the Product Rule.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field>

A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response.

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea1`, `value`); ANSWER := ans1; if evalb(ANSWER-eval(parse(StudentResponse))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mn>60</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>204</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>324</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>308</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>150</mn></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mn>60</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>204</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>324</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>308</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>150</mn></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): n3 := Generate(integer(range=2..3)): r2 := Generate(choose([-10..-1,1..10])): fn5 := sort(Generate(polynom(integer(range=r2),x,degree=n3))): n4 := Generate(integer(range=2..3)): fn6 := sort(Generate(polynom(integer(range=r2),x,degree=n4))): fn7 := fn5/fn6; ans2 := simplify(diff(fn7,x));

KiYsKCokSSJ4RzYiIiIjIiIlRiUiIioiIzUiIiJGKywqKiRGJSIiJEYuRiQiIidGJUYvRipGKyEiIg==

LCQqJiwsKiRJInhHNiIiIiUiIicqJEYmIiIkIiNGKiRGJiIiIyIjZ0YmIiM/ISM6IiIiRjIsKkYqRitGLUYpRiZGKSIjNUYyISIjRjU=

JSFH

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 11.2</Text-field>

Usando a regra apropriada de derivadas, encontrar a derivada de :

f (x) = `$fn7`

JSFH

<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

JSFHans2;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb(simplify(`$ANSWER`-`$RESPONSE`)=0);

JSFH

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

Use the Quotient Rule.

Replace this text with hint 2.

Replace this text with hint 3.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea2`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the Quotient Rule.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea2`, `value`); ANSWER := ans2; if evalb(simplify(ANSWER-eval(parse(StudentResponse)))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>27</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>60</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>20</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>15</mn></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>10</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>27</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>60</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>20</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>15</mn></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>10</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): nn1 := Generate(integer(range=1..5)): nn2 := Generate(integer(range=1..5)): fn[1] := sort(Generate(polynom(integer(range=-20..20),x,degree=nn1)),x): hn[1] := `Sum, Difference, Identity, Power, Constant Multiple`: fn[2] := sort(Generate(polynom(integer(range=-20..20),x,degree=nn2)),x): hn[2] := `Sum, Difference, Identity, Power, Constant Multiple`: fn[3] := fn[1]*fn[2]: hn[3] := `Sum, Difference, Identity, Power, Constant Multiple, Product`: fn[4] := fn[1]/fn[2]: hn[4] := `Sum, Difference, Identity, Power, Constant Multiple, Quotient`: fn[5] := fn[2]/fn[1]: hn[5] := `Sum, Difference, Identity, Power, Constant Multiple, Quotient`: nn := Generate(choose([1,2,3,4,5])): fn1 := fn[nn]; hn1 := hn[nn];

KiYsJkkieEc2IiEiJCEjNyIiIkYoLCwqJEYkIiIlISIoKiRGJCIiJCEiIyokRiQiIiNGKEYkISImISM6RighIiI=

SWhuU3VtLH5EaWZmZXJlbmNlLH5JZGVudGl0eSx+UG93ZXIsfkNvbnN0YW50fk11bHRpcGxlLH5RdW90aWVudEc2Ig==

JSFH

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 11.3</Text-field>

Usando a regra apropriada de derivadas, encontrar a derivada de

f(x) = `$fn1`

JSFH

<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

simplify(diff(fn1,x));

JSFH

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb(`$ANSWER`-`$RESPONSE`=0);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

Use the following differentiation rules: `$hn1`.

Replace this text with hint 2.

Replace this text with hint 3.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea0`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the following differentiation rules: </mtext><mtext></mtext><mrow><mi>Sum, Difference, Identity, Power, Constant Multiple, Quotient</mi></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea0`, `value`); ANSWER := simplify(diff(fn1,x)); if evalb(ANSWER-eval(parse(StudentResponse))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mn>21</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>116</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>23</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>8</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>5</mn></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>15</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mn>21</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>116</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>23</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>8</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>5</mn></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>15</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" layout="Heading 1">Estendendo a Regra de Pot\303\252ncia para Expoentes Inteiros Negativos </Text-field>

A Regra do Quociente pode ser aplicada para estender a Regra de Pot\303\252ncia a expoentes inteiros negativos.

Seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= um inteiro negativo eLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYjNiYtSSNtaUdGJDYlUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RJSZuZTtGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUkjbW5HRiQ2JFEiMEYnRi9GLy1GSTYjUSFGJ0Yv. A deriva\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUSJuRidGNUY4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrL0Y5USdub3JtYWxGJw== pelas Regras do Quociente e de Pot\303\252ncia segue depois de observar que 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 e que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUYsNiVRIm5GJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= \303\251 um inteiro positivo. Tratando a derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUSJuRidGNUY4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrL0Y5USdub3JtYWxGJw== como a derivada de um quociente, leva a:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYsNi1RMSZJbnZpc2libGVUaW1lcztGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUkmbWZyYWNHRiQ2KC1GIzYsLUkjbWlHRiQ2I1EhRictRiM2KUZOLUklbXN1cEdGJDYlLUZPNiVRInVGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JS1GLDYtUSgmbWludXM7RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1GTzYlUSJuRidGWkZnbkYvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZGLUZKNigtRk82JVEiZEYnRlpGZ24tRk82JVEjZHVGJ0ZaRmduLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZhcC8lKWJldmVsbGVkR0Y0RkYtSSNtbkdGJDYkRl5wRi9GL0ZGLUZncDYkUSsmdW1pbnVzMDsxRidGL0ZGRmRvRk4tSShtZmVuY2VkR0YkNiQtRiM2JkZORlRGTkYvRi9GTkYvLUYjNiZGTi1GVTYlRlxxLUZncDYkUSIyRidGL0Zhb0ZORi9GXHBGX3BGYnBGZHBGTkYv

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYsNi1RMSZJbnZpc2libGVUaW1lcztGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0ZGLUklbXN1cEdGJDYlLUZKNiVRInVGJ0ZNRlAtRiM2Ky1GLDYtUSgmbWludXM7RidGL0YyRjVGN0Y5RjtGPUY/RkFGREZJRlotSSNtbkdGJDYkUSIxRidGLy1GLDYtUScmcGx1cztGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUZobjYkUSIyRidGL0ZGRklGLy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRko2I1EhRidGLw==

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

Portanto, a f\303\263rmula na Regra de Pot\303\252ncia se aplica quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for qualquer inteiro -- positivo, zero, ou negativo.

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): r3 := Generate(choose([-10..-1,1..10])): n5 := Generate(integer(range=r3)): n6 := Generate(integer(range=-8..-1)): fn8 := n5 * x ^ n6; ans4 := simplify(diff(fn8,x));

LCQqJEkieEc2IiEiKEYm

LCQqJEkieEc2IiEiKSIjXA==

JSFH

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 11.4</Text-field>

Usando as regras de deriva\303\247\303\243o apropriadas, encontrar a derivada de: f(x) = `$fn8`

JSFH

<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

JSFHans4;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb(`$ANSWER`-`$RESPONSE`=0);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

Use the Power Rule.

Replace this text with hint 2.

Replace this text with hint 3.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea3`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the Power Rule.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea3`, `value`); ANSWER := ans4; if evalb(ANSWER-eval(parse(StudentResponse))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mn>49</mn></mrow><mrow><msup><mi>x</mi><mn>8</mn></msup></mrow></mfrac></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mn>49</mn></mrow><mrow><msup><mi>x</mi><mn>8</mn></msup></mrow></mfrac></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): fn9 := 0: fn10 := 0: while (fn9=0) do r4 := Generate(choose([-10..-1,1..10])): n7 := Generate(integer(range=r4)): n8 := Generate(integer(range=-8..-1)): r5 := Generate(choose([-10..-1,1..10])): n9 := Generate(integer(range=r5)): n10 := Generate(integer(range=-8..-1)): fn9 := n7*x^n8 + n9*x^n10: end do: while (fn10=0) do r6 := Generate(choose([-10..-1,1..10])): n11 := Generate(integer(range=r6)): n12 := Generate(integer(range=-8..-1)): r7 := Generate(choose([-10..-1,1..10])): n13 := Generate(integer(range=r7)): n14 := Generate(integer(range=-8..-1)): fn10 := n11*x^n12 + n13*x^n14: end do: fn11 := Generate(choose([fn9,fn10,fn9*fn10,fn9/fn10,fn10/fn9])); ans5 := sort(simplify(diff(fn11,x)));

LCQqJkkieEc2IiEiKCwmKiRGJCEiIiIiJyokRiQhIiQiIiRGKSEjOQ==

LCQqKCwmKiRJInhHNiIiIiMiIiQiIiJGKkYqLCZGJUYoRipGKiEiI0YmISImIyIjY0Yp

JSFH

<Text-field style="Heading 1" layout="Heading 1">Question 11.5</Text-field>

Usando as regras de deriva\303\247\303\243o apropriadas, encontrar a derivada de: f(x) = `$fn11`

JSFH

<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

JSFHans5;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb(`$ANSWER`-`$RESPONSE`=0);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

Use the Power Rule.

Does the Product Rule need to be used?

Does the Quotient Rule need to be used?

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea4`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the Power Rule.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea4`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Does the Product Rule need to be used?</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea4`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Does the Quotient Rule need to be used?</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea4`, `value`); ANSWER := ans5; if evalb(ANSWER-eval(parse(StudentResponse))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mn>56</mn><mn>3</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfrac><mrow><mrow><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>5</mn></msup></mrow></mfrac></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mn>56</mn><mn>3</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfrac><mrow><mrow><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>5</mn></msup></mrow></mfrac></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" layout="Heading 1">Resumo</Text-field>

As Regras de Deriva\303\247\303\243o fornecem as ferramentas necess\303\241rias para derivar uma fun\303\247\303\243o sem avaliar explicitamente um limite. Com estas regras \303\251 poss\303\255vel derivar qualquer polin\303\264mio ou fun\303\247\303\243o racional. As Regras do Produto e do Quociente s\303\243o particularmentehe dignas de nota porque elas n\303\243o fazem paralelo com suas correspondentes Regras de Limite. J\303\241 vimos que a Regra de Pot\303\252ncia, de fato, se aplica para qualquer inteiro.

A habilidade para reconhecer e aplicar apropriadamente regras de deriva\303\247\303\243o \303\251 essencial para o sucesso no restante deste curso. Voc\303\252 precisa ser capaz de computar derivadas com facilidade, rapidez e precis\303\243o.

<Text-field style="Heading 1" layout="Heading 1">Pr\303\263ximo Cap\303\255tulo</Text-field>

O maplet Differentiation (ou o Tutor M\303\251todos de Diferencia\303\247\303\243o) deve ser usado para desenvolver a habilidade em selecionar a regra de deriva\303\247\303\243o apropriada para cada passo em um problema de deriva\303\247\303\243o.

Nossa lista de Regras de Deriva\303\247\303\243o n\303\243o est\303\241 completa ainda. A Regra da Cadeia \303\251 t\303\243o importante -- e poderosa -- que \303\251 discutida em um cap\303\255tulo em separado. Nossa lista de regras ficar\303\241 completa quando aprendermos sobre deriva\303\247\303\243o de fun\303\247\303\265es trigonom\303\251tricas no Cap\303\255tulo 13. Ambos os cap\303\255tulos fornecer\303\243o experi\303\252ncia adicional no c\303\241lculo de derivadas.

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