Chapter 13.6, Problem 92ES

Prove: If $x=x\left(t\right)\text{and}y=y\left(t\right)$ are differentiable at t, and if $z=f\left(x,y\right)$ is differentiable at the point $\left(x\left(t\right),y\left(t\right)\right),$ then

$\frac{dz}{dt}=\nabla z\cdot \text{r'}\left(t\right)$

where $\text{r}\left(t\right)=x\left(t\right)\text{i}+y\left(t\right)\text{j}\text{.}$