Questão 2. ( 2,5 pontos) Considere a função
f(u, v)=e^{v} \operatorname{sen} u
Sabendo que u=u(r, s)=e^{s} sen r e v=v(r, s)=e^{r} sen s, determine as derivadas parciais \frac{\partial f}{\partial r} e \frac{\partial f}{\partial s} quando r=s=0.

Passo 1

O enunciado define
f(u,v)=e^{v}\sin u,
com
u(r,s)=e^{s}\sin r, \qquad v(r,s)=e^{r}\sin s,
e pede as derivadas parciais \frac{\partial f}{\partial r} e \frac{\partial f}{\partial s} avaliadas em r=s=0.
Para isso, vamos usar a regra da cadeia em duas variáveis:
\frac{\partial f}{\partial r} = f_{u}\,\frac{\partial u}{\partial r} + f_{v}\,\frac{\partial v}{\partial r}, \quad \frac{\partial f}{\partial s} = f_{u}\,\frac{\partial u}{\partial s} + f_{v}\,\frac{\partial v}{\partial s}.

Passo 2

Calcular as derivadas parciais de f, u e v:
  1. Derivadas de f:
f_{u}=\frac{\partial}{\partial u}\bigl(e^{v}\sin u\bigr) =e^{v}\cos u, \qquad f_{v}=\frac{\partial}{\partial v}\bigl(e^{v}\sin u\bigr) =e^{v}\sin u.
  1. Derivadas de u(r,s):
\frac{\partial u}{\partial r} =\frac{\partial}{\partial r}\bigl(e^{s}\sin r\bigr) =e^{s}\cos r, \qquad \frac{\partial u}{\partial s} =\frac{\partial}{\partial s}\bigl(e^{s}\sin r\bigr) =e^{s}\sin r.
  1. Derivadas de v(r,s):
\frac{\partial v}{\partial r} =\frac{\partial}{\partial r}\bigl(e^{r}\sin s\bigr) =e^{r}\sin s, \qquad \frac{\partial v}{\partial s} =\frac{\partial}{\partial s}\bigl(e^{r}\sin s\bigr) =e^{r}\cos s.

Passo 3

Avaliar tudo em r=0, s=0:
  • u(0,0)=e^{0}\sin0=0, v(0,0)=e^{0}\sin0=0.
  • f_{u}(0,0)=e^{v(0,0)}\cos\bigl(u(0,0)\bigr)=e^{0}\cos0=1.
  • f_{v}(0,0)=e^{0}\sin0=0.
  • \frac{\partial u}{\partial r}(0,0)=e^{0}\cos0=1.
  • \frac{\partial u}{\partial s}(0,0)=e^{0}\sin0=0.
  • \frac{\partial v}{\partial r}(0,0)=e^{0}\sin0=0.
  • \frac{\partial v}{\partial s}(0,0)=e^{0}\cos0=1.
Agora apliquei a regra da cadeia:
\frac{\partial f}{\partial r}(0,0) =1\cdot1 + 0\cdot0 = 1, \quad \frac{\partial f}{\partial s}(0,0) =1\cdot0 + 0\cdot1 = 0.

Resposta

\frac{\partial f}{\partial r}(0,0)=1, \qquad \frac{\partial f}{\partial s}(0,0)=0.

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