Derivatives of Inverse Functions


Suppose that we know all about a function `f` and its derivative `f'`. If `f` has an inverse, `g`, can we use our knowledge of `f` to compute the derivative of `g`? Yes!
If `f` and `g` are inverse functions, then [`g'(x) = 1/(f'(g(x)))`]
In the applet above, we will see a geometric justification for this formula. Drag the slider through the steps and consider the questions below.


Alternate Proof

If `f` and `g` are inverse functions and `x` is in the domain of `g`, then [`f(g(x)) = x`] Take the derivative of both sides, using the chain rule on the left: [`f'(g(x))g'(x) = 1`] Solve for `g'(x)`: [`g'(x) = 1/(f'(g(x)))`] Done!