If $\phi = f(x,y)$ where $x=t^2, y=\sin(t^2)$, express $\frac{d\phi}{dt}$ in terms of $f_{x},f_{y}$ and $t$.

Question

I'm very confused how to go about answering this sort of question. Any help would be much appreciated!

Answer 1

Use the chain rule: $$\frac{df(x(t),y(t))}{dt} = \frac{\partial f(x(t),y(t))}{\partial x}\frac{dx(t)}{dt} + \frac{\partial f(x(t),y(t))}{\partial y}\frac{dy(t)}{dt}\ .$$ Note: I know the notation I used is ugly and suboptimal, but I wanted to make explicit the dependence on the variables.

Answer 2

The multivariate chain rule yields, with your notations: $$\frac{\mathrm d\phi}{\mathrm d t}=f_x\frac{\mathrm d x}{\mathrm d t}+f_y\frac{\mathrm d y}{\mathrm d t}.$$