Relating the derivatives of an injective function to the derivatives of it's inverse function

Question

Suppose $f : \mathbb{R} \to \mathbb{R}$ an injective function that is differentiable. Being injective; it has an inverse function $f^{-1}$. How are the derivatives of the inverse function related to the derivatives of the original function? For instance, I know that for nice enough $f$ where $f(x) = y$ we have:

$$(f^{-1}){'}(y) = \lim_{y \to y_0} \frac{(f^{-1})(y) - (f^{-1})(y_0)}{y - y_0} = \lim_{x \to x_0} \frac{x - x_0}{f(x) - f(x_0)} = \frac{1}{f'(x)}$$

For nice functions, are higher order derivatives of the inverse function (such as $(f^{-1}){'}{'}(y)$) related in nice ways to the derivatives of the ordinary function?

Answer 1

Repeated application of your formula shows: $$ \mathrm D^2 f^{-1}(y) = \mathrm D \frac{1}{f' \circ f^{-1}(y)} = -(f' \circ f^{-1}(y))^{-2} f'' \circ f^{-1}(y) \frac{1}{f' \circ f^{-1}(y)} = -\frac{f'' \circ f^{-1}(y)}{(f' \circ f^{-1}(y))^3} $$ The formula quickly gets complicated and impractical.