Utilize a Regra da Cadeia para determinar as derivadas parciais indicadas.
z=x^{2}+x y^{3}, \quad x=u w^{2}+w^{3}, \quad y=u+v e^{w}; \frac{\partial z}{\partial u}, \frac{\partial z}{\partial v}, \frac{\partial z}{\partial w} \quad quando u=2, v=1, w=0

Passo 1

O enunciado define
z = x^{2} + x\,y^{3},\quad x = u\,w^{2} + w^{3},\quad y = u + v\,e^{w}
e pede \dfrac{\partial z}{\partial u},\,\dfrac{\partial z}{\partial v},\,\dfrac{\partial z}{\partial w} no ponto u=2,\;v=1,\;w=0.
Vamos usar a Regra da Cadeia para funções de várias variáveis:
\frac{\partial z}{\partial u} = z_{x}\,\frac{\partial x}{\partial u} + z_{y}\,\frac{\partial y}{\partial u},
e análogos para \partial z/\partial v e \partial z/\partial w.

Passo 2

Calcular as derivadas parciais de z em função de x,y:
z_{x} = \frac{\partial}{\partial x}\bigl(x^{2} + x\,y^{3}\bigr) = 2x + y^{3},
z_{y} = \frac{\partial}{\partial y}\bigl(x^{2} + x\,y^{3}\bigr) = 3\,x\,y^{2}.

Passo 3

Calcular as derivadas parciais de x e y em função de u,v,w:
\frac{\partial x}{\partial u} = w^{2},\quad \frac{\partial x}{\partial v} = 0,\quad \frac{\partial x}{\partial w} = 2\,u\,w + 3\,w^{2},
\frac{\partial y}{\partial u} = 1,\quad \frac{\partial y}{\partial v} = e^{w},\quad \frac{\partial y}{\partial w} = v\,e^{w}.

Passo 4

Montar as fórmulas gerais:
\frac{\partial z}{\partial u} = (2x + y^{3})\,w^{2} + (3\,x\,y^{2})\cdot 1,
\frac{\partial z}{\partial v} = (2x + y^{3})\cdot 0 + (3\,x\,y^{2})\,e^{w},
\frac{\partial z}{\partial w} = (2x + y^{3})\,(2\,u\,w + 3\,w^{2}) + (3\,x\,y^{2})\,(v\,e^{w}).

Passo 5

Avaliar em u=2,\;v=1,\;w=0. Primeiro,
x = 2\cdot 0^{2} + 0^{3} = 0, \quad y = 2 + 1\cdot e^{0} = 3, \quad e^{w}=1.
Substituindo em cada derivada:
\frac{\partial z}{\partial u} = (2\cdot 0 + 3^{3})\cdot 0^{2} + 3\cdot 0\cdot 3^{2} = 27\cdot 0 + 0 = 0,
\frac{\partial z}{\partial v} = 3\cdot 0\cdot 3^{2}\cdot 1 = 0,
\frac{\partial z}{\partial w} = (2\cdot 0 + 27)\cdot(2\cdot 2\cdot 0 +3\cdot0^{2}) +3\cdot0\cdot3^{2}\cdot1 =27\cdot0+0=0.

Resposta

\displaystyle \frac{\partial z}{\partial u}=0,\quad \frac{\partial z}{\partial v}=0,\quad \frac{\partial z}{\partial w}=0 em u=2,\;v=1,\;w=0.

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