Let $w=f(x,y)$, where $x$ and $y$ are functions of $t$. Then, according to the chain rule, $\dfrac{dw}{dt}$ is equal to
- $\dfrac{dw}{dx} \dfrac{dx}{dt} + \dfrac{dw}{dy} \dfrac{dt}{dt} \\$
- $\dfrac{\partial w}{\partial x} \dfrac{\partial x}{\partial t} + \dfrac{\partial w}{ \partial y} \dfrac{\partial y}{ \partial t} \\$
- $\dfrac{\partial w}{\partial x} \dfrac{d x}{dt} + \dfrac{\partial w}{ \partial y} \dfrac{dy}{dt} \\$
- $\dfrac{d w}{dx} \dfrac{\partial x}{\partial t} + \dfrac{dw}{dy} \dfrac{\partial y}{ \partial t}$