differentiation of inverse without having inverse

Question

Suppose $y=f(x)$. Calcuate $$\frac{d}{dy} \frac{1}{x}.$$

My effort: If we had the inverse $x=f^{-1}(y)$, then we could do $$\frac{d}{dy} \frac{1}{x}= \frac{d}{dy} \frac{1}{f^{-1}(y)}$$ But, how do we do it if we don't have $f^{-1}(y)$?

Answer 1

Power rule, chain rule:

$$\frac{d}{dy}\frac{1}{x} = \frac{-1}{x^2}\frac{dx}{dy} = \frac{-1}{x^2dy/dx} = \frac{-1}{x^2f'(x)}$$