Question
Prove: If $x=x(t)$ and $y=y(t)$ are differentiable at $t,$ and if $z=f(x, y)$ is differentiable at the point $(x(t), y(t)),$ then$$\frac{d z}{d t}=\nabla z \cdot \mathbf{r}^{\prime}(t)$$where $\mathbf{r}(t)=x(t) \mathbf{i}+y(t) \mathbf{j}$
Step 1
By the chain rule, we have $$ \frac{d z}{d t}=\frac{\partial z}{\partial x} \frac{d x}{d t}+\frac{\partial z}{\partial y} \frac{d y}{d t} $$ Show more…
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