Derivatives of Inverse Functions

If $y=f(x)$ is one-one and onto (so it is invertible), its inverse function $x=f^{-1}(y)$ exists, and — as the two illustrations below suggest — the derivative of the inverse turns out to be closely tied to the derivative of the original function.

Example 1. Let $f(x)=2x-2$, so its inverse is $f^{-1}(x)=\dfrac{x+2}{2}$; write $g(x)=f^{-1}(x)$. Differentiating each: $\dfrac{d}{dx}[f(x)]=2$ and $\dfrac{d}{dx}[g(x)]=\dfrac12$. These two derivatives are reciprocals of one another. …