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	$f^{\prime }(g(x_0))g^{\prime }(x_0)$ (this is the chain rule for differentiation)
2	$f^{″}(g(x_0))(g^{\prime }(x_0){)}^2+f^{\prime }(g(x_0))g^{″}(x_0)$ (obtained by using the chain rule for differentiation twice and using the product rule for differentiation)