Calcule as derivadas partial zpartial u e partial zpartial v.
z=x tan y, x=u^2, y=1 /(1+v);

Passo 1

O enunciado dá
z = x tan y, x = u^2, y = 11+v,
e pede as derivadas parciais partial zpartial u e partial zpartial v. Vamos usar a regra da cadeia para funções compostas.

Passo 2

Para calcular partial zpartial u aplicamos
partial zpartial u = partial zpartial x,partial xpartial u + partial zpartial y,partial ypartial u
Agora:
  • partial zpartial x = tan y,
  • partial xpartial u = 2u,
  • partial zpartial y = x,sec^2 y,
  • partial ypartial u = 0 (pois y não depende de u).
    Logo
partial zpartial u = (tan y),(2u) + (x,sec^2 y)cdot 0 = 2u,tan y
Substituindo y=11+v, fica
partial zpartial u = 2u,tan!l(11+vr)

Passo 3

Para calcular partial zpartial v aplicamos
partial zpartial v = partial zpartial x,partial xpartial v + partial zpartial y,partial ypartial v
Agora:
  • partial xpartial v = 0 (pois x=u^2 independe de v),
  • partial ypartial v = -1(1+v)^2.
    Logo
partial zpartial v = (tan y)cdot 0 + l(x,sec^2 yr),l(-1(1+v)^2r) = -x,sec^2 y(1+v)^2
Substituindo x=u^2 e y=11+v, obtemos
partial zpartial v = -u^2,sec^2!l(tfrac11+vr)(1+v)^2

Resposta

partial zpartial u = 2u,tan!l(11+vr), partial zpartial v = -u^2,sec^2!l(11+vr)(1+v)^2

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