Escrever a regra da cadeia para

c) h(u, v, w) = f(x(u, v, w), y(u, v), z(w))

Passo 1

Bom, pessoal, calcular a derivada parcial em relação a u :

∂ h ∂ u = ∂ f ∂ x ⋅ ∂ x ∂ u + ∂ f ∂ y ⋅ ∂ y ∂ u + ∂ f ∂ z ⋅ ∂ z ∂ u

∂ h ∂ u = ∂ f ∂ x ⋅ ∂ x ∂ u + ∂ f ∂ y ⋅ ∂ y ∂ u + ∂ f ∂ z ⋅ 0

∂ h ∂ u = ∂ f ∂ x ⋅ ∂ x ∂ u + ∂ f ∂ y ⋅ ∂ y ∂ u

Passo 2

Agora, calculando a derivada parcial em relação a v :

∂ h ∂ v = ∂ f ∂ x ⋅ ∂ x ∂ v + ∂ f ∂ y ⋅ ∂ y ∂ v + ∂ f ∂ z ⋅ ∂ z ∂ v

∂ h ∂ v = ∂ f ∂ x ⋅ ∂ x ∂ v + ∂ f ∂ y ⋅ ∂ y ∂ v + ∂ f ∂ z ⋅ 0

∂ h ∂ v = ∂ f ∂ x ⋅ ∂ x ∂ v + ∂ f ∂ y ⋅ ∂ y ∂ v

Passo 3

Por fim, calculando a derivada parcial em relação a w :

∂ h ∂ w = ∂ f ∂ x ⋅ ∂ x ∂ w + ∂ f ∂ y ⋅ ∂ y ∂ w + ∂ f ∂ z ⋅ ∂ z ∂ w

∂ h ∂ w = ∂ f ∂ x ⋅ ∂ x ∂ w + ∂ f ∂ y ⋅ 0 + ∂ f ∂ z ⋅ ∂ z ∂ w

∂ h ∂ w = ∂ f ∂ x ⋅ ∂ x ∂ w + ∂ f ∂ z ⋅ ∂ z ∂ w

Resposta

As derivadas são iguais a:

∂ h ∂ u = ∂ f ∂ x ⋅ ∂ x ∂ u + ∂ f ∂ y ⋅ ∂ y ∂ u

∂ h ∂ v = ∂ f ∂ x ⋅ ∂ x ∂ v + ∂ f ∂ y ⋅ ∂ y ∂ v

∂ h ∂ w = ∂ f ∂ x ⋅ ∂ x ∂ w + ∂ f ∂ z ⋅ ∂ z ∂ w

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