O gradiente de f em (x, y, z)=(2,1,0).
\begin{aligned} \frac{\partial f}{\partial x}(x, y, z) & =\frac{\partial}{\partial x}(x+x y+x y z) \\\ & =\frac{\partial}{\partial x}(x)+y \frac{\partial}{\partial x}(x)+y z \frac{\partial}{\partial x}(x) \\\ & =1+y+y z \\\ \frac{\partial f}{\partial y}(x, y, z) & =\frac{\partial}{\partial y}(x+x y+x y z) \\\ & =x \frac{\partial}{\partial y}(1)+x \frac{\partial}{\partial y}(y)+x z \frac{\partial}{\partial y}(y) \\\ & =0+x+x z=x+x z \end{aligned}
\begin{aligned} \frac{\partial f}{\partial z}(x, y, z) & =\frac{\partial}{\partial z}(x+x y+x y z) \\\ & =x \frac{\partial}{\partial z}(1)+x y \frac{\partial}{\partial z}(1)+x y \frac{\partial}{\partial z}(z) \\\ & =0+0+x y=x y \end{aligned}
Portanto \nabla f(x, y, z)=(1+y+y z, x+x z, x y).
Logo o gradiente de f em ( 2,1,0 ) é
\nabla f(2,1,0)=(1+1+1.0,2+2.0,2.1)=(2,2,2)

Passo 1

O enunciado pede para calcular o gradiente de
f(x,y,z)=x + x\,y + x\,y\,z
no ponto (x,y,z)=(2,1,0). Vamos usar derivadas parciais para obter
\displaystyle \nabla f=(\tfrac{\partial f}{\partial x},\tfrac{\partial f}{\partial y},\tfrac{\partial f}{\partial z})

Passo 2

Calculemos \displaystyle \frac{\partial f}{\partial x}. Temos
\frac{\partial f}{\partial x} =\frac{\partial}{\partial x}(x)+\frac{\partial}{\partial x}(x\,y)+\frac{\partial}{\partial x}(x\,y\,z) =1 + y + y\,z.

Passo 3

Agora \displaystyle \frac{\partial f}{\partial y}. Note que x e z são constantes em relação a y:
\frac{\partial f}{\partial y} =\frac{\partial}{\partial y}(x)+\frac{\partial}{\partial y}(x\,y)+\frac{\partial}{\partial y}(x\,y\,z) =0 + x + x\,z = x + x\,z.

Passo 4

Finalmente \displaystyle \frac{\partial f}{\partial z}. Aqui só o termo x\,y\,z depende de z:
\frac{\partial f}{\partial z} =\frac{\partial}{\partial z}(x)+\frac{\partial}{\partial z}(x\,y)+\frac{\partial}{\partial z}(x\,y\,z) =0 + 0 + x\,y = x\,y.

Passo 5

Reunindo,
\nabla f(x,y,z)=(1 + y + y\,z,\;x + x\,z,\;x\,y).
Para (x,y,z)=(2,1,0):
1 + 1 + 1\cdot0 = 2,\quad 2 + 2\cdot0 = 2,\quad 2\cdot1 = 2.

Resposta

\nabla f(2,1,0) = (2,2,2).

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