When is the derivative of an inverse function equal to the reciprocal of the derivative? When is this statement true?
$$\dfrac {\mathrm dx}{\mathrm dy} = \frac 1 {\frac {\mathrm dy}{\mathrm dx}}$$
where $y=y(x)$. I think that $y(x)$ has to be bijective in order to have an inverse and let the expression $\dfrac {\mathrm dx}{\mathrm dy}$ make sense. But is there any other condition?

Assume $g(f(x))=x$. Then
$$g'(f(x))f'(x)=1$$ and then
$$g'(f(x))=\frac1{f'(x)}$$
Note that we need also that $f'(x)\neq 0$. All the conditions (the injectivity and the differentability of $f$ and that $f'$ does not vanish) must meet in a neighbourhood of the point where you are differentiating, that is, this works locally.
See the inverse function theorem.
