Para cada item a seguir, calcule \partial z / \partial s e \partial z / \partial t.
z=\frac{\operatorname{sen}(\theta)}{r}, r=s t, \theta=s^{2}+t^{2}

Passo 1

O enunciado define
z = \frac{\operatorname{sen}(\theta)}{r}, \quad r = s\,t, \quad \theta = s^{2} + t^{2},
e pede \frac{\partial z}{\partial s} e \frac{\partial z}{\partial t}.
Vamos usar a regra da cadeia em função de duas variáveis intermediárias r e \theta.

Passo 2

Calcular as derivadas parciais de z em relação a r e a \theta:
\frac{\partial z}{\partial r} = \frac{\partial}{\partial r}\Bigl(\operatorname{sen}(\theta)\,r^{-1}\Bigr) = -\,\frac{\operatorname{sen}(\theta)}{r^{2}},
\frac{\partial z}{\partial \theta} = \frac{\partial}{\partial \theta}\Bigl(\operatorname{sen}(\theta)\,r^{-1}\Bigr) = \frac{\cos(\theta)}{r}.

Passo 3

Calcular as derivadas parciais de r e \theta em relação a s e t:
\frac{\partial r}{\partial s} = t, \quad \frac{\partial r}{\partial t} = s,
\frac{\partial \theta}{\partial s} = 2s, \quad \frac{\partial \theta}{\partial t} = 2t.

Passo 4

Aplicar a regra da cadeia para \displaystyle \frac{\partial z}{\partial s}:
\frac{\partial z}{\partial s} = \frac{\partial z}{\partial r}\,\frac{\partial r}{\partial s} + \frac{\partial z}{\partial \theta}\,\frac{\partial \theta}{\partial s} = \Bigl(-\frac{\operatorname{sen}(\theta)}{r^{2}}\Bigr)\,t + \Bigl(\frac{\cos(\theta)}{r}\Bigr)\,2s.
Substituindo r = s\,t e \theta = s^{2}+t^{2}, obtemos
\frac{\partial z}{\partial s} = -\frac{t\,\operatorname{sen}(s^{2}+t^{2})}{(s\,t)^{2}} + \frac{2s\,\cos(s^{2}+t^{2})}{s\,t} = -\frac{\operatorname{sen}(s^{2}+t^{2})}{s^{2}\,t} + \frac{2\,\cos(s^{2}+t^{2})}{t}.

Passo 5

Aplicar a regra da cadeia para \displaystyle \frac{\partial z}{\partial t}:
\frac{\partial z}{\partial t} = \frac{\partial z}{\partial r}\,\frac{\partial r}{\partial t} + \frac{\partial z}{\partial \theta}\,\frac{\partial \theta}{\partial t} = \Bigl(-\frac{\operatorname{sen}(\theta)}{r^{2}}\Bigr)\,s + \Bigl(\frac{\cos(\theta)}{r}\Bigr)\,2t.
Substituindo r = s\,t e \theta = s^{2}+t^{2}, obtemos
\frac{\partial z}{\partial t} = -\frac{s\,\operatorname{sen}(s^{2}+t^{2})}{(s\,t)^{2}} + \frac{2t\,\cos(s^{2}+t^{2})}{s\,t} = -\frac{\operatorname{sen}(s^{2}+t^{2})}{s\,t^{2}} + \frac{2\,\cos(s^{2}+t^{2})}{s}.

Resposta

\frac{\partial z}{\partial s} = -\frac{\operatorname{sen}(s^{2}+t^{2})}{s^{2}\,t} + \frac{2\,\cos(s^{2}+t^{2})}{t}, \quad \frac{\partial z}{\partial t} = -\frac{\operatorname{sen}(s^{2}+t^{2})}{s\,t^{2}} + \frac{2\,\cos(s^{2}+t^{2})}{s}.

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