Utilize a Regra da Cadeia para determinar as derivadas parciais indicadas.
u=\sqrt{r^{2}+s^{2}}, r=y+x \cos t, \quad s=x+y \operatorname{sen} t ;
\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial t} quando x=1, \quad y=2, \quad t=0

Passo 1

O enunciado define
u=\sqrt{r^{2}+s^{2}},\quad r=y+x\cos t,\quad s=x+y\operatorname{sen} t
e pede \displaystyle \frac{\partial u}{\partial x}, \displaystyle \frac{\partial u}{\partial y} e \displaystyle \frac{\partial u}{\partial t} no ponto x=1, y=2, t=0. Vamos usar a Regra da Cadeia para funções compostas.

Passo 2

Primeiro, calculamos as derivadas parciais de u em relação a r e a s, e as de r e s em relação a x,y,t:
u_r=\frac{\partial}{\partial r}\sqrt{r^2+s^2}=\frac{r}{\sqrt{r^2+s^2}},\qquad u_s=\frac{s}{\sqrt{r^2+s^2}},
r_x=\cos t,\quad r_y=1,\quad r_t=-x\operatorname{sen} t,
s_x=1,\quad s_y=\operatorname{sen} t,\quad s_t=y\cos t.

Passo 3

Aplicando a Regra da Cadeia:
\frac{\partial u}{\partial x} = u_r\,r_x + u_s\,s_x =\frac{r\cos t + s}{\sqrt{r^2+s^2}},
\frac{\partial u}{\partial y} = u_r\,r_y + u_s\,s_y =\frac{r + s\operatorname{sen} t}{\sqrt{r^2+s^2}},
\frac{\partial u}{\partial t} = u_r\,r_t + u_s\,s_t =\frac{-r\,x\operatorname{sen} t + s\,y\cos t}{\sqrt{r^2+s^2}}.

Passo 4

No ponto x=1, y=2, t=0 temos \cos0=1, \sen0=0,
r=2 + 1\cdot1 = 3,\quad s=1 + 2\cdot0 = 1,\quad u=\sqrt{3^2+1^2}=\sqrt{10}.
Substituindo nas fórmulas:
\frac{\partial u}{\partial x} =\frac{3\cdot1 + 1}{\sqrt{10}}=\frac{4}{\sqrt{10}}, \quad \frac{\partial u}{\partial y} =\frac{3 + 1\cdot0}{\sqrt{10}}=\frac{3}{\sqrt{10}},
\frac{\partial u}{\partial t} =\frac{-3\cdot1\cdot0 + 1\cdot2\cdot1}{\sqrt{10}} =\frac{2}{\sqrt{10}}.

Resposta

\frac{\partial u}{\partial x}\Big|_{(1,2,0)}=\frac{4}{\sqrt{10}},\quad \frac{\partial u}{\partial y}\Big|_{(1,2,0)}=\frac{3}{\sqrt{10}},\quad \frac{\partial u}{\partial t}\Big|_{(1,2,0)}=\frac{2}{\sqrt{10}}.

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