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

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Guia de de Estudo de C\303\241lculo

\302\251 Maplesoft, a division of Waterloo Maple Inc., 2005

Cap\303\255tulo 11:Regras de Deriva\303\247\303\243o

Avalia\303\247\303\243o de derivadas (derivatives) usando a defini\303\247\303\243o depende da habilidade para avaliar (evaluate) o limite (limit) do quociente da diferen\303\247a (difference quotient). Isto \303\251 tedioso e demorado. E, pior, os limites podem ser extremamente dif\303\255ceis de serem avaliados (evaluate). As regras de deriva\303\247\303\243o (differentiation) s\303\243o uma cole\303\247\303\243o de regras gerais para computar derivadas formadas de polin\303\264mios, pot\303\252ncias, raizes e m\303\272ltiplos de constante (constant), somas, diferen\303\247as, produtos, quocientes ( quotients) desses tipos de fun\303\247\303\265es (functions). Composi\303\247\303\265es de fun\303\247\303\265es (functions) s\303\243o manipuladas separadamente no pr\303\263ximo cap\303\255tulo, em regra de cadeia (chain rule).

Tenha cuidado para n\303\243o confundir leis 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 Differentiation maplet \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 o(a) iniciar\303\241 no uso dessa ferramenta.

<Text-field style="Heading 1" layout="Heading 1">Cap\303\255tulo Gloss\303\241rio</Text-field>

Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio de Matem\303\241tica Maple ( Maple Math Dictionary).

assumption

chainrule

constant

continuous

derivative

differencequotient

difference

differentiable

differentiate

differentiation

evaluate

even

explicit

exponent

function

graph

limit

linear

midpoint

multiple

negative

numerator

origin

polynomial

positive

powerrule

productrule

product

proof

quotientrule

quotient

rateofchange

rationalfunction

remainder

slope

variable

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

restart;

with(Student[Calculus1]):

<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 (Product ), as regras n\303\243o s\303\243o t\303\243o \303\263bvias e s\303\243o bastante diferentes das correspondentes leis de limites.

Nome da Regra

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

Condi\303\247\303\265es

Constante(Constant)

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

Identidade(Identity)

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

Pot\303\252ncia(Power)

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 um inteiro positivo

M\303\272ltiplo de Constante(Constant Multiple)

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

Soma(Sum)

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

Diferen\303\247\303\243(Difference)

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

Produto(Product)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2KEYrLUYjNiYtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y9USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZWLUYwNiQtRiM2JC1GLDYlUSJ4RidGOUY8RkNGQ0ZDRistRiM2Ji1GLDYlUSJnRidGOUY8Rj9GWUZDRitGQ0ZDLUZANi1RIidGJ0ZDRkVGSEZKRkxGTkZQRlJGVEZXLUZANi1RMSZJbnZpc2libGVUaW1lcztGJ0ZDRkVGSEZKRkxGTkZQRlJGVEZXLUZANi1RIj1GJ0ZDRkVGSEZKRkxGTkZQRlJGVEZXRmJvRjRGXG9GX29GWUZiby1GQDYtUScmcGx1cztGJ0ZDRkVGSEZKRkxGTkZQRlJGVEZXRmJvRitGam5GNkZib0Zfb0ZZRitGQw==

Quociente(Quotient)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtSShtZmVuY2VkR0YkNiQtRiM2JkYrLUkmbWZyYWNHRiQ2KC1GIzYnLUYsNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GQ1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk0vJSlzdHJldGNoeUdGTS8lKnN5bW1ldHJpY0dGTS8lKGxhcmdlb3BHRk0vJS5tb3ZhYmxlbGltaXRzR0ZNLyUnYWNjZW50R0ZNLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGZm4tRjM2JC1GIzYkLUYsNiVRInhGJ0Y/RkJGSUZJRitGSS1GIzYmRistRiM2Ji1GLDYlUSJnRidGP0ZCRkVGaW5GSUYrRkkvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlxwLyUpYmV2ZWxsZWRHRk1GK0ZJRkktRiM2JUYrLUZGNi1RKCZwcmltZTtGJ0ZJRktGTkZQRlJGVEZWRlgvRmVuUSwwLjExMTExMTFlbUYnRmduRkkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EjLTFGJy1GRjYtUSI9RidGSUZLRk5GUEZSRlRGVkZYRlpGZ24tRkY2LVExJkludmlzaWJsZVRpbWVzO0YnRklGS0ZORlBGUkZURlZGWEZaRmduLUY4NigtRiM2MUYrRmJvRjxGXnEtRkY2LVEiJ0YnRklGS0ZORlBGUkZURlZGWEZaRmduRmluRl5xLUZGNi1RKCZtaW51cztGJ0ZJRktGTkZQRlJGVEZWRlhGWkZnbkZecS1GIzYmRjxGRUZpbkZJRmRvRmVxRmluRitGSS1GIzYoRistRjA2JUZkby1JI21uR0YkNiRRIjJGJ0ZJL0ZpcFEiMEYnRitGaW5GK0ZJRmdvRmpvRl1wRl9wRitGSQ==

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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 M\303\272tiplo de 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

\303\211 muito dif\303\255cil dar uma prova (proof) 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 gerador de gr\303\241fico Maple, em outros casos, o Maple pode ser usado para identificar o quociente da diferen\303\247a como derivadas. Porem, deve-se ter muito cuidado 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 (graph) de uma fun\303\247\303\243o constante \303\251 uma reta horizontal, i.e., tem inclina\303\247\303\243o (slope) 0. (Veja Figura 11.1.)

plot(2.3, x=-3..3);

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

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 intu\303\255da do nosso conhecimento de fun\303\247\303\265es lineares (linear). O gr\303\241fico da fun\303\247\303\243o 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 \303\251 uma linha reta que passa pela origem (origin) com inclina\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . assim, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEkZn4nRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtRiM2Ji1GPTYtUSIoRidGQC9GQ0Y4RkUvRkhGOEZJRktGTUZPL0ZSUSwwLjE2NjY2NjdlbUYnL0ZVRmhuLUYjNiQtRiw2JVEieEYnRjZGOUZALUY9Ni1RIilGJ0ZARmVuRkVGZm5GSUZLRk1GT0Znbi9GVVEsMC4xMTExMTExZW1GJ0ZARitGQC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmhvLUkjbW5HRiQ2JFEiMUYnRkBGQEYrRkA= .

Uma prova rigorosa baseada na defini\303\247\303\243o tambem n\303\243o \303\251 dif\303\255cil. Desde que sendo a fun\303\247\303\243o identidade 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,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 de Pot\303\252ncia (Power Rule) com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= \303\251 a Regra da Identidade, a saber,

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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIwRicvRjtRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRkFGK0ZB = 1

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

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

for i from 1 to 10 do (x+h)^i = expand( (x+h)^i ) end do;

LywmSSJ4RzYiIiIiSSJoR0YlRiZGIw==

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIiNGKCwoKiQpRiZGKkYoRigqKEYqRihGJkYoRilGKEYoKiQpRilGKkYoRig=

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIiRGKCwqKiQpRiZGKkYoRigqKEYqRigpRiYiIiNGKEYpRihGKCooRipGKEYmRigpRilGMEYoRigqJClGKUYqRihGKA==

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIiVGKCwsKiQpRiZGKkYoRigqKEYqRigpRiYiIiRGKEYpRihGKCooIiInRigpRiYiIiNGKClGKUY0RihGKCooRipGKEYmRigpRilGMEYoRigqJClGKUYqRihGKA==

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIiZGKCwuKiQpRiZGKkYoRigqKEYqRigpRiYiIiVGKEYpRihGKCooIiM1RigpRiYiIiRGKClGKSIiI0YoRigqKEYyRigpRiZGNkYoKUYpRjRGKEYoKihGKkYoRiZGKClGKUYwRihGKCokKUYpRipGKEYo

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIidGKCwwKiQpRiZGKkYoRigqKEYqRigpRiYiIiZGKEYpRihGKCooIiM6RigpRiYiIiVGKClGKSIiI0YoRigqKCIjP0YoKUYmIiIkRigpRilGOkYoRigqKEYyRigpRiZGNkYoKUYpRjRGKEYoKihGKkYoRiZGKClGKUYwRihGKCokKUYpRipGKEYo

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIihGKCwyKiQpRiZGKkYoRigqKEYqRigpRiYiIidGKEYpRihGKCooIiNARigpRiYiIiZGKClGKSIiI0YoRigqKCIjTkYoKUYmIiIlRigpRikiIiRGKEYoKihGOEYoKUYmRjxGKClGKUY6RihGKCooRjJGKClGJkY2RigpRilGNEYoRigqKEYqRihGJkYoKUYpRjBGKEYoKiQpRilGKkYoRig=

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIilGKCw0KiQpRiZGKkYoRigqKEYqRigpRiYiIihGKEYpRihGKCooIiNHRigpRiYiIidGKClGKSIiI0YoRigqKCIjY0YoKUYmIiImRigpRikiIiRGKEYoKigiI3FGKClGJiIiJUYoKUYpRkBGKEYoKihGOEYoKUYmRjxGKClGKUY6RihGKCooRjJGKClGJkY2RigpRilGNEYoRigqKEYqRihGJkYoKUYpRjBGKEYoKiQpRilGKkYoRig=

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIipGKCw2KiQpRiZGKkYoRigqKEYqRigpRiYiIilGKEYpRihGKCooIiNPRigpRiYiIihGKClGKSIiI0YoRigqKCIjJSlGKClGJiIiJ0YoKUYpIiIkRihGKCooIiRFIkYoKUYmIiImRigpRikiIiVGKEYoKihGPkYoKUYmRkJGKClGKUZARihGKCooRjhGKClGJkY8RigpRilGOkYoRigqKEYyRigpRiZGNkYoKUYpRjRGKEYoKihGKkYoRiZGKClGKUYwRihGKCokKUYpRipGKEYo

LyokKSwmSSJ4RzYiIiIiSSJoR0YnRigiIzVGKCw4KiQpRiZGKkYoRigqKEYqRigpRiYiIipGKEYpRihGKCooIiNYRigpRiYiIilGKClGKSIiI0YoRigqKCIkPyJGKClGJiIiKEYoKUYpIiIkRihGKCooIiQ1I0YoKUYmIiInRigpRikiIiVGKEYoKigiJF8jRigpRiYiIiZGKClGKUZGRihGKCooRj5GKClGJkZCRigpRilGQEYoRigqKEY4RigpRiZGPEYoKUYpRjpGKEYoKihGMkYoKUYmRjZGKClGKUY0RihGKCooRipGKEYmRigpRilGMEYoRigqJClGKUYqRihGKA==

Em particular, note que 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 . (Dizemos que 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 quando 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 existe; em particular, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNictRiw2JVEiT0YnL0Y1RkJGPi1GOzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGaG4tRiM2Ji1GOzYtUSIoRidGPi9GQUY2RkMvRkZGNkZHRklGS0ZNL0ZQUSwwLjE2NjY2NjdlbUYnL0ZTRmJvLUYjNiQtRiw2JVEiaEYnRjRGN0Y+LUY7Ni1RIilGJ0Y+Rl9vRkNGYG9GR0ZJRktGTUZhby9GU1EsMC4xMTExMTExZW1GJ0Y+RitGPkYrRj5GK0Y+ significa 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 .) 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 (exponent) \303\251 zero, um inteiro negativo (negative) , um n\303\272mero racional (rational) , 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 M\303\272ltiplo de Constante </Text-field>

A Regra M\303\272ltiplo de Constante deve ser explorada intuitivamente antes de ser provada. Assim, sendo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiYtRiw2JVEnJiM5MTY7RicvRjVGQkY+LUY7Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGaG4tRiw2JVEieEYnRjRGN0Y+RitGPkYrRj4= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiYtRiw2JVEnJiM5MTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y5LyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUC1GLDYlUSJ5RicvRjdRJXRydWVGJy9GOlEnaXRhbGljRidGOS1GPTYtUSI9RidGOUZARkJGREZGRkhGSkZML0ZPUSwwLjI3Nzc3NzhlbUYnL0ZSRmhuLUYjNihGKy1GIzYnLUYsNiVRImZGJ0ZWRlgtRj02LVEwJkFwcGx5RnVuY3Rpb247RidGOUZARkJGREZGRkhGSkZMRk5GUS1GIzYmLUY9Ni1RIihGJ0Y5L0ZBRldGQi9GRUZXRkZGSEZKRkwvRk9RLDAuMTY2NjY2N2VtRicvRlJGXHAtRiM2JkYrLUYjNiYtRiw2JVEieEYnRlZGWC1GPTYtUSIrRidGOUZARkJGREZGRkhGSkZML0ZPUSwwLjIyMjIyMjJlbUYnL0ZSRmlwLUYsNiVRImhGJ0ZWRlhGOUYrRjktRj02LVEiKUYnRjlGaW9GQkZqb0ZGRkhGSkZMRltwL0ZSUSwwLjExMTExMTFlbUYnRjlGK0Y5LUY9Ni1RKCZtaW51cztGJ0Y5RkBGQkZERkZGSEZKRkxGaHBGanAtRiM2J0Zeb0Zhby1GIzYmRmZvLUYjNiRGYnBGOUZecUY5RitGOUYrRjlGK0Y5RitGOQ== a abscissa e ordenada (the run and rise), respectivamente, para f. Se 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 , ent\303\243o a ordenada (the rise) para F ser\303\241

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 = 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 = 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJyYjOTE2O0YnL0Y1RkJGPkY6LUYsNiVRInlGJ0Y0RjdGPkYrRj4=

Ent\303\243o, 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 para todol 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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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNictRiw2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUYjNiYtRj02LVEiKEYnRkAvRkNGOEZFL0ZIRjhGSUZLRk1GTy9GUlEsMC4xNjY2NjY3ZW1GJy9GVUZobi1GIzYkLUYsNiVRInhGJ0Y2RjlGQC1GPTYtUSIpRidGQEZlbkZFRmZuRklGS0ZNRk9GZ24vRlVRLDAuMTExMTExMWVtRidGQEYrRkAtRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yNzc3Nzc4ZW1GJy9GVUZoby1GIzYnLUYsNiVRImtGJ0Y2RjktRj02LVExJkludmlzaWJsZVRpbWVzO0YnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtRiM2Jy1GLDYlUSJmRidGNkY5RjxGVkYrRkBGK0ZARitGQEYrRkA=, 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 (rate of change) 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, M\303\272ltiplo de Constante e da Soma.

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

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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 (Product Rule) 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 por\303\251m

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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 no numerador

(numerator), tal que quocientes da diferen\303\247a 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 (assumption) de que f \303\251 diferenci\303\241vel em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= implica que f \303\251 cont\303\255nua (continuous) 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 (Quotient Rule) \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 da diferen\303\247a envolvendo f e g podem 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 deriva\303\247\303\243o e avalia\303\247\303\243o. Dos tr\303\252s, somente no primeiro exemplo \303\251

ainda razo\303\241vel tentar avaliar a derivada por meio da defini\303\247\303\243o.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Example 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 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

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

Remarks

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

derivada original (original derivative)

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

Regras de Soma e Diferen\303\247a

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy1JKG1mZW5jZWRHRiQ2JC1GIzYmRistSSVtc3VwR0YkNiUtRiw2JVEieEYnRjVGOC1JI21uR0YkNiRRIjNGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGWEZYLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRlgvJSZmZW5jZUdGSC8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZbcEYvRistRko2JC1GIzYnRlQtRmhuNi1RMSZJbnZpc2libGVUaW1lcztGJ0ZYRltvRl1vRl9vRmFvRmNvRmVvRmdvRmlvRlxwLUZPNiVGUS1GVTYkUSIyRidGWEZaRitGWEZYLUZobjYtUScmcGx1cztGJ0ZYRltvRl1vRl9vRmFvRmNvRmVvRmdvRmlvRlxwRi9GKy1GSjYkLUYjNiRGUUZYRlhGK0ZY

Regra da Constante aplicada

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

Regra de M\303\272ltiplo de 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNilGKy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSy1JJm1mcmFjR0YkNigtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y4USdpdGFsaWNGJy1GLDYlUSNkeEYnRlRGVy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGW28vJSliZXZlbGxlZEdGPEYrLUYjNiYtRjQ2LVEiKEYnRjcvRjtGVkY9L0ZARlZGQUZDRkVGRy9GSlEsMC4xNjY2NjY3ZW1GJy9GTUZoby1GIzYrRistRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJ0ZURlctSSNtbkdGJDYkUSIzRidGNy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y3LUY0Ni1RKCZtaW51cztGJ0Y3RjpGPUY/RkFGQ0ZFRkcvRkpRLDAuMjIyMjIyMmVtRicvRk1GX3EtRiM2J0ZkcEYzLUYjNiZGKy1GX3A2JUZhcC1GZXA2JFEiMkYnRjdGaHBGK0Y3RitGNy1GNDYtUSIrRidGN0Y6Rj1GP0ZBRkNGRUZHRl5xRmBxRmFwRmpxRmRwRjctRjQ2LVEiKUYnRjdGZW9GPUZmb0ZBRkNGRUZHRmdvL0ZNUSwwLjExMTExMTFlbUYnRjdGK0Y3LUY0Ni1RIj1GJ0Y3RjpGPUY/RkFGQ0ZFRkcvRkpRLDAuMjc3Nzc3OGVtRicvRk1GZnItRiM2KUYrRmFxRltxLUYjNiYtRmVwNiRRIjZGJ0Y3RjNGYXBGN0ZqcS1GZXA2JFEiMUYnRjdGN0YrRjdGK0Y3

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

q1 := x^3-3*x^2+x+3: Diff( q1, x ) = diff( q1, x );

\303\211 aconselh\303\241vel repetir o exemplo no Differentiation maplet, equivalente a construir no Tutor de Deriva\303\247\303\243o Passo-a-Passo (Step-byStep Differentiation Tutor) dispon\303\255vel no menu Ferramentas/Tutores/C\303\241lculo-Uma Vari\303\241vel (Tools/Tutors/Calculus-SingleVariable). O chamado M\303\251todo de Deriva\303\247\303\243o (Differentiation Methods) nesse menu pode ser iniciado para a fun\303\247\303\243o do Exemple 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 de Deriva\303\247\303\243o Passo-a-Passo aplicado a 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

Pressione o bot\303\243o Start, 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)(Rule Definition), Ajuda (help), e outro modo de aplicar regras (Apply Rule). 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 "understood", 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="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KUYrLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbW9HRiQ2LlEwJkRpZmZlcmVudGlhbEQ7RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGPC8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZQL0Y+USVib2xkRicvJStmb250d2VpZ2h0R0ZULUYjNidGKy1GIzYmRjYtRiw2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y+USxib2xkLWl0YWxpY0YnRlVGU0ZVRitGU0ZVLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Ziby8lKWJldmVsbGVkR0Y8LUY3Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y9RkBGQkZERkZGSEZKRkxGTkZRLUYjNictRjc2LlEiKEYnRjpGPS9GQUZqbkZCL0ZFRmpuRkZGSEZKRkwvRk9RLDAuMTY2NjY2N2VtRicvRlJGYnAtRjI2KC1GIzYnRistRiM2J0Zlbi1GNzYuUSgmbWludXM7RidGOkY9RkBGQkZERkZGSEZKRkwvRk9RLDAuMjIyMjIyMmVtRicvRlJGXnEtSSNtbkdGJDYlUSIxRidGU0ZVRlNGVUYrRlNGVS1GIzYnRistRiM2KEYrLUYjNidGKy1JJW1zdXBHRiQ2JUZlbi1GYXE2JVEiMkYnRlNGVS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0ZTRlUtRjc2LlEiK0YnRjpGPUZARkJGREZGRkhGSkZMRl1xRl9xRl1yRlNGVUYrRlNGVUZdb0Zgb0Zjb0Zlby1GNzYuUSIpRidGOkY9Rl9wRkJGYHBGRkZIRkpGTEZhcC9GUlEsMC4xMTExMTExZW1GJ0ZTRlVGK0ZTRlVGK0ZTRlU=">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</Equation> </Text-field>

Esta derivada ser\303\241 computada 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

q2 := (t-1)/(t^2+2): Diff( q2, t ) = diff( q2, t ); `` = simplify( diff(q2,t) );

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemple 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="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">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</Equation></Text-field>

\303\240 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 de Deriva\303\247\303\243o Passo-a-Passo n\303\243o acredita que a regra do Produto possa ser aplicada aqui. A dica sugere que a Regra da Cadeia (Chain Rule) deveria ser aplicada. Como n\303\243o conhecemos ainda a Regra da Cadeia, tem-se que 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 termwise 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

q3 := (x^4+3*x^2+1)^2: Diff( q3, x ) = expand( diff(q3,x) );

A deriva\303\247\303\243o obtida quando a fun\303\247\303\243o \303\251 expandida antes de ser derivada (differentiated) termwise \303\251

Diff( q3, x ) = Diff( expand(q3), x ); `` = diff( expand(q3), x );

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 termwise 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 ser\303\241 endere\303\247ado ao cap\303\255tulo Regra da Cadeia ( Chain Rule ).

Quest\303\243o 11.1 Usando a regra apropriada da derivada, encontrar a derivada de:

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

Quest\303\243o 11.2 Usando a regra apropriada da derivada, encontrar a derivada de:

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

Quest\303\243o 11.3 Usando a regra apropriada da derivada , encontrar a derivada de:

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

<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.

Sendo 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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

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

Quest\303\243o 11.4 Usando a regra apropriada da derivada , encontrar a derivada de:

Quest\303\243o 11.5 Usando a regra apropriada da derivada , encontrar a derivada de:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIj1GJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVi1JJm1mcmFjR0YkNigtRiM2KC1GQzYtUSomdW1pbnVzMDtGJ0ZARkZGSEZKRkxGTkZQRlIvRlVRLDAuMjIyMjIyMmVtRicvRlhGXG8tRlo2KC1JI21uR0YkNiRRIjVGJ0ZALUYjNiUtSSVtc3VwR0YkNiVGOi1GIzYlLUZhbzYkUSI2RidGQEYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRi9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZnAvJSliZXZlbGxlZEdGPy1GQzYtUSIrRidGQEZGRkhGSkZMRk5GUEZSRltvRl1vLUZaNigtRmFvNiRRIjNGJ0ZALUYjNiVGOkYvRjJGYXBGZHBGZ3BGaXBGPUZALUYjNiYtRlo2KC1GYW82JFEiNEYnRkAtRiM2JS1GZ282JUY6LUYjNiUtRmFvNiRRIjhGJ0ZARi9GMkZecEYvRjJGYXBGZHBGZ3BGaXBGW3EtRlo2KEZbcC1GIzYlLUZnbzYlRjotRiM2JS1GYW82JFEiMkYnRkBGL0YyRl5wRi9GMkZhcEZkcEZncEZpcEZARmFwRmRwRmdwRmlwRj1GQA==

<Text-field style="Heading 1" layout="Heading 1">Cap\303\255tulo 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 (explicitly ) um limite. Com estas regras \303\251 poss\303\255vel derivar qualquer polin\303\264mio polynomial 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 Leis 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. \303\211 preciso ser capaz de computar derivadas com facilidade, rapidez e precis\303\243o.

<Text-field style="Heading 1" layout="Heading 1">Qual \303\251 o Pr\303\263ximo?</Text-field>

O Differentiation maplet, (ou Tutor de Deriva\303\247\303\243o Passo-a-Passo)(or Step-by-Step Differentiation Tutor) 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.

Este \303\251 o meio da Unidade 2 (Unit 2); tome algum tempo para rever seus tr\303\252s cap\303\255tulos.

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 destes (of these) cap\303\255tulos fornecer\303\243o experi\303\252ncia adicional em computar derivadas.

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