# Difference between revisions of "Chain rule for higher derivatives"

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 $n$.
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).