Three variables partial derivatives using chain rule

Question

Question:

Suppose $w=F(x,y,t), \ y=g(x,t), \ x=h(t)$. Find $dw/dt$ in terms of partial derivatives of $F,g$, and the derivative of $h$.

So I know that I need to use chain rule. But $w$ is a function of $x,y,t$ and when I do partial derivatives, wouldn't there be $dw/dt$ on both sides of equation?

Answer 1

Take into account that, if you write $w$ as $w = F\left(h(t), g(h(t),t), t\right) = F(A_1,A_2,A_3)$, then:

$$ \frac{\partial w}{\partial t} = \frac{\partial w}{\partial A_1} \frac{\partial A_1}{\partial t} + \frac{\partial w}{\partial A_2} \frac{\partial A_2}{\partial t} + \frac{\partial w}{\partial A_3} \frac{\partial A_3}{\partial t}$$

Note now that:

\begin{align} \partial_t A_1 & = h'(t), \\ \partial_t A_2 & = \partial_t g(h(t),t) = \frac{\partial g}{\partial h} \frac{\partial h}{\partial t} + \frac{\partial g}{\partial t} \frac{\partial t}{\partial t} , \\ \partial_t A_3 & = 1 . \\ \end{align}

Hope this helps.

Cheers!