Answer
$$16$$
Work Step by Step
Given
$$
g(x, y, z, w)=x+2 y+3 z+5 w, \quad \mathbf{r}(t)=\left\langle t^{2}, t^{3}, t, t-2\right\rangle, \quad t=1
$$
since $\mathbf{r}(1)=\langle 1,1,1,-1\rangle$ and
$$
\begin{aligned}
\nabla g &=\left\langle\frac{\partial g}{\partial x}, \frac{\partial g}{\partial y},\frac{\partial g}{\partial z},\frac{\partial g}{\partial w}\right\rangle \\
&=\langle 1,2,3,5 \rangle \\
\mathbf{r}^{\prime}(t) &=\left\langle 2t,3t^2, 1,1\right\rangle
\end{aligned}
$$
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 1,2,3,5 \rangle \cdot\langle 2,3, 1,1 \rangle \\
&= 2+6+3+5=16
\end{aligned}
