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 $x_{0}$
0	$\!f(g(x_{0}))$ (taking the 0th derivative means doing nothing)
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)