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

The chain rule is stated in many versions:

Version type	Statement
specific point, named 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: $\frac{d}{dx}[f(g(x))]{|}_{x=x_0}=f^{\prime }(g(x_0))g^{\prime }(x_0)$
generic point, named functions, point notation	Suppose $f$ and $g$ are functions of one variable. Then, we have $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))g^{\prime }(x)$ wherever the right side expression makes sense.
generic point, named functions, point-free notation	Suppose $f$ and $g$ are functions of one variable. Then, $(f\circ g{)}^{\prime }=(f^{\prime }\circ g)\cdot g^{\prime }$ where the right side expression makes sense, where $\cdot$ denotes the pointwise product of functions.
pure Leibniz notation	Suppose $u=g(x)$ is a function of $x$ and $v=f(u)$ is a function of $u$. Then, $\frac{dv}{dx}=\frac{dv}{du}\frac{du}{dx}$

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).