Answer
$$0$$
Work Step by Step
Given
$$
f(x, y)=x e^{y}, \quad \mathbf{r}(t)=\left\langle t^{2}, t^{2}-4 t\right\rangle, \quad t=0
$$
since $\mathbf{r}(0)=\langle 0,0\rangle$ and
$$
\begin{aligned}
\nabla f &=\left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right\rangle \\
&=\langle e^{y}, x e^{y}\rangle \\
\mathbf{r}^{\prime}(t) &=\left\langle 2t, 2t-4\right\rangle
\end{aligned}
$$
Then
\begin{aligned}
\left.\frac{d}{d t} f(\mathbf{r}(t))\right|_{t=0}&=\nabla f_{\mathbf{r}(0)} \cdot \mathbf{r}^{\prime}(0)\\
&=\langle1,0\rangle \cdot\langle 0,-4\rangle \\
&= 0
\end{aligned}
