Determine:
\frac{d z}{d u}, \frac{d z}{d v}, \frac{d z}{d w} quando u=1, v=2, w=0, onde z=\sqrt{r^{2}+s^{2}}, r=v+u \cos (w), s=u+v \operatorname{sen}(w).

Passo 1

O enunciado nos dá
z = \sqrt{r^2 + s^2}, \quad r = v + u\cos(w), \quad s = u + v\operatorname{sen}(w)
e pede as derivadas parciais \frac{\partial z}{\partial u}, \frac{\partial z}{\partial v} e \frac{\partial z}{\partial w} no ponto u=1, v=2, w=0. Vamos usar a regra da cadeia para funções compostas e depois avaliar em (1,2,0).

Passo 2

Antes de aplicar a regra da cadeia, calculamos as derivadas parciais de r e s em relação a cada variável:
\frac{\partial r}{\partial u} = \cos(w), \quad \frac{\partial r}{\partial v} = 1, \quad \frac{\partial r}{\partial w} = -u\operatorname{sen}(w),
\frac{\partial s}{\partial u} = 1, \quad \frac{\partial s}{\partial v} = \operatorname{sen}(w), \quad \frac{\partial s}{\partial w} = v\cos(w).

Passo 3

A derivada de z=\sqrt{r^2+s^2} em relação a uma variável x\in\{u,v,w\} é, pela regra da cadeia e pela derivada de \sqrt{\cdot}:
\frac{\partial z}{\partial x} = \frac{1}{2\sqrt{r^2+s^2}}\cdot 2\bigl(r\,\frac{\partial r}{\partial x}+s\,\frac{\partial s}{\partial x}\bigr) = \frac{r\,\frac{\partial r}{\partial x}+s\,\frac{\partial s}{\partial x}}{\sqrt{r^2+s^2}}.

Passo 4

Agora avaliamos r, s e as derivadas parciais de r,s no ponto dado u=1, v=2, w=0. Como \cos(0)=1 e \operatorname{sen}(0)=0,
r(1,2,0) = 2 + 1\cdot1 = 3, \quad s(1,2,0) = 1 + 2\cdot0 = 1, \quad \sqrt{r^2+s^2} = \sqrt{3^2+1^2} = \sqrt{10}.
E
\frac{\partial r}{\partial u}\Big|_{(1,2,0)} = 1, \quad \frac{\partial r}{\partial v}\Big|_{(1,2,0)} = 1, \quad \frac{\partial r}{\partial w}\Big|_{(1,2,0)} = -1\cdot0=0,
\frac{\partial s}{\partial u}\Big|_{(1,2,0)} = 1, \quad \frac{\partial s}{\partial v}\Big|_{(1,2,0)} = 0, \quad \frac{\partial s}{\partial w}\Big|_{(1,2,0)} = 2\cdot1=2.

Passo 5

Substituindo na fórmula geral:
\frac{\partial z}{\partial u} = \frac{3\cdot1 + 1\cdot1}{\sqrt{10}} = \frac{4}{\sqrt{10}}, \quad \frac{\partial z}{\partial v} = \frac{3\cdot1 + 1\cdot0}{\sqrt{10}} = \frac{3}{\sqrt{10}},
\frac{\partial z}{\partial w} = \frac{3\cdot0 + 1\cdot2}{\sqrt{10}} = \frac{2}{\sqrt{10}}.

Resposta

\frac{\partial z}{\partial u}\Big|_{(1,2,0)} = \frac{4}{\sqrt{10}}, \quad \frac{\partial z}{\partial v}\Big|_{(1,2,0)} = \frac{3}{\sqrt{10}}, \quad \frac{\partial z}{\partial w}\Big|_{(1,2,0)} = \frac{2}{\sqrt{10}}.

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