Chain rule for higher derivatives: Difference between revisions

Latest revision as of 02:21, 5 December 2023

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)

@@ Line 9: / Line 9: @@
 {| class="sortable" border="1"
 ! Value of <math>n</math> !! Formula for <math>n^{th}</math> derivative of <math>f \circ g</math> at <math>x_0</math>
+|-
+| 0 || <math>\! f(g(x_0)) </math> (taking the 0th derivative means doing nothing)
 |-
 | 1 || <math>\! f'(g(x_0))g'(x_0)</math> (this is the [[chain rule for differentiation]])
 |-
-| 2 || <math>\! f''(g(x_0))(g'(x_0))^2 + f'(g(x_0))g''(x_0)</math> (obtained by using the [[chain rule for differentiation]] twice ''and'' using the [[product rule for differentiation]]).
+| 2 || <math>\! f''(g(x_0))(g'(x_0))^2 + f'(g(x_0))g''(x_0)</math> (obtained by using the [[chain rule for differentiation]] twice ''and'' using the [[product rule for differentiation]])
 |}