1. Usando a regra da cadeia para z=f(x, y) e w=f(x, y, z), calcule: \frac{\partial z}{\partial t}, \frac{\partial z}{\partial s} e \frac{\partial w}{\partial t}, \frac{\partial w}{\partial s} e \frac{\partial w}{\partial r}.
    (j) w=\sqrt{x^{2}+y^{2}+z^{2}}, x=\operatorname{tg}(t), y=\cos (r), z=\operatorname{sen}(s)

Passo 1

O enunciado pede para usar a regra da cadeia em
z=f(x,y),\quad w=f(x,y,z)
no caso específico
w=\sqrt{x^{2}+y^{2}+z^{2}},\quad x=\tan(t),\quad y=\cos(r),\quad z=\sin(s).
Ou seja, primeiro vamos tratar z como função de (t,s) e depois w como função de (t,s,r).

Passo 2

Calcular \frac{\partial z}{\partial t} e \frac{\partial z}{\partial s}.
Como z=\sin(s) não depende de t, temos
\frac{\partial z}{\partial t}=0.
Derivando em relação a s:
\frac{\partial z}{\partial s} =\frac{d}{ds}\bigl(\sin(s)\bigr) =\cos(s).

Passo 3

Agora aplicamos a regra da cadeia para w=\sqrt{x^{2}+y^{2}+z^{2}}
de modo que, para qualquer parâmetro u,
\frac{\partial w}{\partial u} =\frac{1}{2\sqrt{x^{2}+y^{2}+z^{2}}}\;2\Bigl(x\frac{\partial x}{\partial u} + y\frac{\partial y}{\partial u} + z\frac{\partial z}{\partial u}\Bigr) =\frac{x\,x_{u}+y\,y_{u}+z\,z_{u}}{\sqrt{x^{2}+y^{2}+z^{2}}},
onde x_{u}=\partial x/\partial u, etc.
Para u=t:
x=\tan(t)\implies x_{t}=\operatorname{sec}^{2}(t),\quad y_{t}=0,\quad z_{t}=0,
logo
\frac{\partial w}{\partial t} =\frac{x\,\operatorname{sec}^{2}(t)}{\sqrt{x^{2}+y^{2}+z^{2}}} =\frac{\tan(t)\,\operatorname{sec}^{2}(t)}{\sqrt{\tan^{2}(t)+\cos^{2}(r)+\sin^{2}(s)}}.

Passo 4

Para u=s:
x_{s}=0,\quad y_{s}=0,\quad z_{s}=\cos(s),
\frac{\partial w}{\partial s} =\frac{z\,\cos(s)}{\sqrt{x^{2}+y^{2}+z^{2}}} =\frac{\sin(s)\,\cos(s)}{\sqrt{\tan^{2}(t)+\cos^{2}(r)+\sin^{2}(s)}}.
Para u=r:
x_{r}=0,\quad y_{r}=-\sin(r),\quad z_{r}=0,
\frac{\partial w}{\partial r} =\frac{y\,(-\sin(r))}{\sqrt{x^{2}+y^{2}+z^{2}}} =-\frac{\cos(r)\,\sin(r)}{\sqrt{\tan^{2}(t)+\cos^{2}(r)+\sin^{2}(s)}}.

Resposta

\displaystyle \frac{\partial z}{\partial t}=0,\quad \frac{\partial z}{\partial s}=\cos(s), \displaystyle \frac{\partial w}{\partial t} =\frac{\tan(t)\,\operatorname{sec}^{2}(t)}{\sqrt{\tan^{2}(t)+\cos^{2}(r)+\sin^{2}(s)}}, \displaystyle \frac{\partial w}{\partial s} =\frac{\sin(s)\,\cos(s)}{\sqrt{\tan^{2}(t)+\cos^{2}(r)+\sin^{2}(s)}}, \displaystyle \frac{\partial w}{\partial r} =-\frac{\cos(r)\,\sin(r)}{\sqrt{\tan^{2}(t)+\cos^{2}(r)+\sin^{2}(s)}}.

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