Use a regra da cadeia para determinar partial zpartial s e partial zpartial t.
c) z=xy, x=2 s+3 t, y=3 s-2 t

Passo 1

Temos a função z=xy com as relações x=2s+3t e y=3s-2t. O objetivo é achar as derivadas parciais partial zpartial s e partial zpartial t. O conteúdo principal é a regra da cadeia para derivadas parciais, usando as fórmulas gerais
partial zpartial s = partial zpartial xpartial xpartial s + partial zpartial ypartial ypartial s, partial zpartial t = partial zpartial xpartial xpartial t + partial zpartial ypartial ypartial t

Passo 2

Para calcular partial zpartial x e partial zpartial y, reescrevemos z=xy=x,y^-1. Assim
partial zpartial x = 1y, partial zpartial y = -xy^2

Passo 3

Agora achamos as derivadas de x e y em relação a s e t:
partial xpartial s = 2, partial xpartial t = 3, partial ypartial s = 3, partial ypartial t = -2

Passo 4

Aplicando a regra da cadeia para partial zpartial s:
partial zpartial s = 1ycdot2 + l(-xy^2r)cdot3 = 2y - 3xy^2 = 2y-3xy^2
Substituindo x=2s+3t e y=3s-2t temos
2y-3x = 2(3s-2t) - 3(2s+3t) = -13t,
logo
partial zpartial s = -13t(3s-2t)^2

Passo 5

De modo análogo, para partial zpartial t:
partial zpartial t = 1ycdot3 + l(-xy^2r)cdot(-2) = 3y + 2xy^2 = 3y+2xy^2
Substituindo
3y+2x = 3(3s-2t) + 2(2s+3t) = 13s,
obtemos
partial zpartial t = 13s(3s-2t)^2

Resposta

[\n\frac{\partial z}{\partial s} = -\frac{13t}{(3s-2t)^2}\n]\n[\n\frac{\partial z}{\partial t} = \frac{13s}{(3s-2t)^2}\n]

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