Chain rule for higher derivatives

This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
View other differentiation rules

Statement

Suppose $n$ is a natural number, and $f$ and $g$ are functions such that $g$ is $n$ times differentiable at $x = x_0$ and $f$ is $n$ times differentiable at $g(x_0)$. Then, $f \circ g$ is $n$ times differentiable at $x_0$. Further, the value of the $n^{th}$ derivative is given by a complicated formula involving compositions, products, derivatives, evaluations, and sums that depends on $n$.

Particular cases

Value of $n$	Formula for $n^{th}$ derivative of $f \circ g$ at $x_0$
1	$\! f'(g(x_0))g'(x_0)$ (this is the chain rule for differentiation)
2	$\! f''(g(x_0))(g'(x_0))^2 + f'(g(x_0))g''(x_0)$ (obtained by using the chain rule for differentiation twice and using the product rule for differentiation).