Answer
$$0$$
Work Step by Step
Given
$$
g(x, y, z)=x y z^{-1}, \quad \mathbf{r}(t)=\left\langle e^{t}, t, t^{2}\right\rangle, \quad t=1
$$
since $\mathbf{r}(1)=\langle e,1,1\rangle$ and
$$
\begin{aligned}
\nabla g &=\left\langle\frac{\partial g}{\partial x}, \frac{\partial g}{\partial y},\frac{\partial g}{\partial z}\right\rangle \\
&=\langle yz^{-1}, xz^{-1},-xyz^{-2} \rangle \\
\mathbf{r}^{\prime}(t) &=\left\langle e^t, 1,2t\right\rangle
\end{aligned}
$$
Then
\begin{aligned}
\left.\frac{d}{d t} f(\mathbf{r}(t))\right|_{t=1}&=\nabla f_{\mathbf{r}(1)} \cdot \mathbf{r}^{\prime}(1)\\
&=\langle 1,e,-e \rangle \cdot\langle e,1,2 \rangle \\
&=e+e-2e=0
\end{aligned}
