# Chain rule

## For derivatives of composed functions / From Wikipedia, the free encyclopedia

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In calculus, the **chain rule** is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if $h=f\circ g$ is the function such that $h(x)=f(g(x))$ for every x, then the chain rule is, in Lagrange's notation,
$h'(x)=f'(g(x))g'(x).$
or, equivalently,
$h'=(f\circ g)'=(f'\circ g)\cdot g'.$

The chain rule may also be expressed in Leibniz's notation. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are dependent variables), then z depends on x as well, via the intermediate variable y. In this case, the chain rule is expressed as ${\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},$ and $\left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},$ for indicating at which points the derivatives have to be evaluated.

In integration, the counterpart to the chain rule is the substitution rule.