Questão 11. Calcular as derivadas parciais das seguintes funções:
k(x, y, z, w, s)=\frac{x y z w s}{x^{2}+x w s^{3}+3 y}

Passo 1

O enunciado pede para calcular as derivadas parciais de
k(x,y,z,w,s) = \frac{x\,y\,z\,w\,s}{x^{2} + x\,w\,s^{3} + 3\,y}.
O conteúdo principal é a regra do quociente em várias variáveis, combinada com regra do produto para o numerador.

Passo 2

Definimos
N(x,y,z,w,s) = x\,y\,z\,w\,s, \qquad D(x,y,y,z,w,s) = x^{2} + x\,w\,s^{3} + 3\,y,
de modo que
k = \frac{N}{D}.

Passo 3

Pela regra do quociente, para qualquer variável u\in\{x,y,z,w,s\}:
\frac{\partial k}{\partial u} = \frac{D\,\frac{\partial N}{\partial u} \;-\; N\,\frac{\partial D}{\partial u}}{D^{2}}.

Passo 4

Para u = x:
\frac{\partial N}{\partial x} = y\,z\,w\,s, \qquad \frac{\partial D}{\partial x} = 2x + w\,s^{3}.
Logo,
\frac{\partial k}{\partial x} = \frac{\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)\,(y\,z\,w\,s) \;-\;(x\,y\,z\,w\,s)\,(2x + w\,s^{3})} {\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)^{2}}.

Passo 5

Para u = y:
\frac{\partial N}{\partial y} = x\,z\,w\,s, \qquad \frac{\partial D}{\partial y} = 3.
Portanto,
\frac{\partial k}{\partial y} = \frac{\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)\,(x\,z\,w\,s) \;-\;(x\,y\,z\,w\,s)\,\cdot 3} {\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)^{2}}.

Passo 6

Para u = z:
\frac{\partial N}{\partial z} = x\,y\,w\,s, \qquad \frac{\partial D}{\partial z} = 0.
Assim,
\frac{\partial k}{\partial z} = \frac{\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)\,(x\,y\,w\,s)} {\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)^{2}} = \frac{x\,y\,w\,s}{x^{2}+x\,w\,s^{3}+3y}.

Passo 7

Para u = w:
\frac{\partial N}{\partial w} = x\,y\,z\,s, \qquad \frac{\partial D}{\partial w} = x\,s^{3}.
Logo
\frac{\partial k}{\partial w} = \frac{\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)\,(x\,y\,z\,s) \;-\;(x\,y\,z\,w\,s)\,(x\,s^{3})} {\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)^{2}}.

Passo 8

Para u = s:
\frac{\partial N}{\partial s} = x\,y\,z\,w, \qquad \frac{\partial D}{\partial s} = 3\,x\,w\,s^{2}.
Assim
\frac{\partial k}{\partial s} = \frac{\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)\,(x\,y\,z\,w) \;-\;(x\,y\,z\,w\,s)\,(3\,x\,w\,s^{2})} {\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)^{2}}.

Resposta

\frac{\partial k}{\partial x} = \frac{\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)\,(y\,z\,w\,s) - x\,y\,z\,w\,s\,(2x + w\,s^{3})} {\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)^{2}},
\frac{\partial k}{\partial y} = \frac{\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)\,(x\,z\,w\,s) - 3\,x\,y\,z\,w\,s} {\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)^{2}},
\frac{\partial k}{\partial z} = \frac{x\,y\,w\,s}{x^{2}+x\,w\,s^{3}+3y},
\frac{\partial k}{\partial w} = \frac{\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)\,(x\,y\,z\,s) - x\,y\,z\,w\,s\,(x\,s^{3})} {\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)^{2}},
\frac{\partial k}{\partial s} = \frac{\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)\,(x\,y\,z\,w) - 3\,x\,y\,z\,w\,s\,(x\,w\,s^{2})} {\bigl(x^{2}+x\,w\,s^{3}+3y\bigr)^{2}}.

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