Calcule partial wpartial x e partial wpartial y nos seguintes casos:
w=u^2+v^3 ; u=3 x-y e v=x+2 y

Passo 1

O enunciado nos dá
w = u^2 + v^3, u = 3x - y, v = x + 2y
e pede as derivadas parciais partial w/partial x e partial w/partial y. Para isso vamos usar a regra da cadeia em funções de duas variáveis.

Passo 2

Primeiro calculamos as derivadas parciais intermediárias:
partial wpartial u = 2u, partial wpartial v = 3v^2,
partial upartial x = 3, partial upartial y = -1, partial vpartial x = 1, partial vpartial y = 2

Passo 3

Pela regra da cadeia temos
partial wpartial x = partial wpartial u,partial upartial x + partial wpartial v,partial vpartial x = 2ucdot3 + 3v^2cdot1 = 6u + 3v^2,
partial wpartial y = partial wpartial u,partial upartial y + partial wpartial v,partial vpartial y = 2ucdot(-1) + 3v^2cdot2 = -2u + 6v^2

Passo 4

Agora substituímos u = 3x - y e v = x + 2y:
partial wpartial x = 6(3x - y) + 3(x + 2y)^2,
partial wpartial y = -2(3x - y) + 6(x + 2y)^2
Opcionalmente expandindo em termos de x,y:
partial wpartial x = 18x - 6y + 3l(x^2 + 4xy + 4y^2r) = 3x^2 + 12xy + 12y^2 + 18x - 6y,
partial wpartial y = -6x + 2y + 6l(x^2 + 4xy + 4y^2r) = 6x^2 + 24xy + 24y^2 - 6x + 2y

Resposta

partial wpartial x = 6(3x - y) + 3(x + 2y)^2, partial wpartial y = -2(3x - y) + 6(x + 2y)^2

Exercícios relacionados