Derivative of two independent varibles.

Question

Consider three quantities related by: $$y=f(t), x=g(t)$$ Then what is $(\frac{\partial y}{\partial x})_t$? Is it simply that we cannot perform such an operation or is it $0$ or $\infty$? If it is one of the latter cases then what is: $(\frac{\partial x}{\partial y})_t$ and is this equal to $[(\frac{\partial y}{\partial x})_t]^{-1}$?

Answer 1

You could use the chain rule:

$(\frac{\partial y}{\partial x})=(\frac{\partial y}{\partial t})(\frac{\partial t}{\partial x})={f'(t)\over g'(t)}$

and it is equal to $(\frac{\partial x}{\partial y})^{-1}$