Exercício 3) Calcule:
x partial zpartial x+y partial zpartial y. Sendo que z=sen(yx).

Passo 1

O enunciado pede o cálculo de
xpartial zpartial x+ypartial zpartial y
onde z=sinl(yxr).
Para isso, vamos usar a regra da cadeia para obter partial zpartial x e partial zpartial y.

Passo 2

Calcular partial zpartial x:
Definindo u=yx, temos z=sin(u), então
partial zpartial x =dsin(u)ducdotpartial upartial x =cos(u)cdotl(-yx^2r) =-yx^2cos!l(yxr)

Passo 3

Calcular partial zpartial y:
Com u=yx, obtemos
partial zpartial y =dsin(u)ducdotpartial upartial y =cos(u)cdot1x =1xcos!l(yxr)

Passo 4

Agora montamos
xpartial zpartial x+ypartial zpartial y =xl(-yx^2cos!tfracyxr) +yl(1xcos!tfracyxr) =-yxcos!tfracyx+yxcos!tfracyx =0

Resposta

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