Use a Regra da Cadeia para achar partial z / partial s e partial z / partial t.
z=x^2 y^3, x=x cos t, y=s sen t

Passo 1

O enunciado diz que
z = x^2 y^3, x = scos t, y = ssin t
e pede partial z/partial s e partial z/partial t usando a regra da cadeia. Vamos precisar de partial z/partial x, partial z/partial y e das derivadas de x e y em relação a s e t.

Passo 2

Calculemos primeiro as derivadas parciais de z em relação a x e y:
partial zpartial x = 2x,y^3, partial zpartial y = 3x^2,y^2
Agora as derivadas de x(s,t) e y(s,t):
partial xpartial s = cos t, partial xpartial t = -ssin t; partial ypartial s = sin t, partial ypartial t = scos t

Passo 3

Aplicando a regra da cadeia para partial z/partial s:
partial zpartial s = partial zpartial x,partial xpartial s + partial zpartial y,partial ypartial s = 2x,y^3cos t ;+; 3x^2,y^2sin t

Passo 4

Agora partial z/partial t:
partial zpartial t = partial zpartial x,partial xpartial t + partial zpartial y,partial ypartial t = 2x,y^3(-ssin t) ;+; 3x^2,y^2(scos t) = -2s,x,y^3sin t ;+; 3s,x^2,y^2cos t

Passo 5

Se quisermos as respostas em termos de s e t, substituímos x = scos t e y = ssin t:
partial zpartial s = 2(scos t)(ssin t)^3cos t + 3(scos t)^2(ssin t)^2sin t = 5,s^4cos^2 t,sin^3 t,
partial zpartial t = -2s,(scos t)(ssin t)^3sin t + 3s,(scos t)^2(ssin t)^2cos t = s^5l(-2cos t,sin^4 t + 3cos^3 t,sin^2 tr)

Resposta

partial zpartial s = 5,s^4cos^2 t,sin^3 t, partial zpartial t = s^5l(-2cos t,sin^4 t + 3cos^3 t,sin^2 tr)

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