Use a regra da cadeia para achar \frac{\partial z}{\partial s} ou \frac{\partial z}{\partial t}
(a) z=x^{2}+x y^{3}, \quad x=u v^{2}+w^{3}, \quad y=u+v e^{w};\n \frac{\partial z}{\partial u}, \frac{\partial z}{\partial v}, \frac{\partial z}{\partial w} quando u=2, v=1, w=0

Passo 1

O enunciado pede as derivadas parciais \frac{\partial z}{\partial u}, \frac{\partial z}{\partial v} e \frac{\partial z}{\partial w} de
z = x^2 + x\,y^3, \quad x = u\,v^2 + w^3, \quad y = u + v\,e^w
avaliadas em (u,v,w)=(2,1,0). Vamos usar a regra da cadeia para funções compostas.

Passo 2

Calculamos as derivadas parciais de z em relação a x e a y:
\frac{\partial z}{\partial x} = 2x + y^3, \qquad \frac{\partial z}{\partial y} = 3x\,y^2.

Passo 3

Calculamos as derivadas parciais de x e y em relação a u, v e w:
\frac{\partial x}{\partial u} = v^2, \quad \frac{\partial x}{\partial v} = 2u\,v, \quad \frac{\partial x}{\partial w} = 3w^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

Pela regra da cadeia temos
\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x}\,\frac{\partial x}{\partial u} + \frac{\partial z}{\partial y}\,\frac{\partial y}{\partial u},
\frac{\partial z}{\partial v} = \frac{\partial z}{\partial x}\,\frac{\partial x}{\partial v} + \frac{\partial z}{\partial y}\,\frac{\partial y}{\partial v},
\frac{\partial z}{\partial w} = \frac{\partial z}{\partial x}\,\frac{\partial x}{\partial w} + \frac{\partial z}{\partial y}\,\frac{\partial y}{\partial w}.

Passo 5

Avaliamo x e y em (u,v,w)=(2,1,0):
x(2,1,0) = 2\cdot1^2 + 0^3 = 2, \qquad y(2,1,0) = 2 + 1\cdot e^0 = 3.

Passo 6

Substituindo tudo:
\frac{\partial z}{\partial u}\biggr|_{(2,1,0)} = (2\cdot2 + 3^3)\cdot1^2 + (3\cdot2\cdot3^2)\cdot1 = (4 + 27) + 54 = 85,
\frac{\partial z}{\partial v}\biggr|_{(2,1,0)} = (4+27)\cdot(2\cdot2\cdot1) + 54\cdot e^0 = 31\cdot4 + 54 = 178,
\frac{\partial z}{\partial w}\biggr|_{(2,1,0)} = (4+27)\cdot0 + 54\cdot(1\cdot e^0) = 54.

Resposta

\frac{\partial z}{\partial u}(2,1,0) = 85, \quad \frac{\partial z}{\partial v}(2,1,0) = 178, \quad \frac{\partial z}{\partial w}(2,1,0) = 54.

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