There is one catch to all the explanations given above where we computed derivatives of inverse functions. To write something like $$\frac {d}{dx}(e^y)=e^y\cdot y'$$ we need to know that the function $y$ has a derivative. The Inverse Function Theorem guarantees this.

It is worth giving one more piece of evidence for the formula above, this time based on increments in function, $\Delta f$, and increments in variable, $\Delta x$. Consider this plot of a function $f$ and its inverse:

Since the graph of the inverse of a function is the reflection of the graph of the function over the line $y=x$, we see that the increments are “switched” when reflected. Hence we see that $$\dfrac {\Delta f^{-1}}{\Delta {x}} = \dfrac {\Delta x}{\Delta f}=\dfrac {1}{\dfrac {\Delta f}{\Delta x}}.$$ Taking the limit as $\Delta x$ goes to $0$, we can obtain the expression for the derivative of $f^{-1}$.
$$\dfrac {d f^{-1}}{d{x}} = \dfrac {1}{\dfrac {d f}{d x}}$$

The inverse function theorem gives us a recipe for computing the derivatives of inverses of functions at points.

If one example is good, two are better:

Finally, let’s see an example where the theorem does not apply.