Chapter 12 - Functions of Several Veriables - 12.5 The Chain Rule - 12.5 Exercises - Page 913: 3

$$ \frac{dw}{dt}=\frac{\partial w}{\partial x} \frac{d x}{dt}+\frac{\partial w}{\partial y} \frac{d y}{dt}+\frac{\partial w}{\partial z} \frac{d z}{dt}. $$

To find $\frac{dw}{dt}$ we apply the chain rule as follows $$ \frac{dw}{dt}=\frac{\partial w}{\partial x} \frac{d x}{dt}+\frac{\partial w}{\partial y} \frac{d y}{dt}+\frac{\partial w}{\partial z} \frac{d z}{dt}. $$