Chain rule for differentiation

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 for two functions

Suppose $f$ and $g$ are functions such that $g$ is differentiable at a point $x=x_0$, and $f$ is differentiable at $g(x_0)$. Then the composite $f\circ g$ is differentiable at $x_0$, and we have:

$\frac{d}{dx}[f(g(x))]{|}_{x=x_0}=f^{\prime }(g(x_0))g^{\prime }(x_0)$

In terms of general expressions:

$\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))g^{\prime }(x)$

In point-free notation, we have:

$(f\circ g{)}^{\prime }=(f^{\prime }\circ g)\cdot g^{\prime }$

where $\cdot$ denotes the pointwise product of functions.

Related rules

• Chain rule for higher derivatives
• Product rule for differentiation
• Product rule for higher derivatives
• Differentiation is linear
• Inverse function theorem (gives formula for derivative of inverse function).