Chain rule for higher derivatives

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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).