Chain rule

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 ${\displaystyle h=f\circ g}$ is the function such that ${\displaystyle h(x)=f(g(x))}$ for every x, then the chain rule is, in Lagrange's notation,

${\displaystyle h'(x)=f'(g(x))g'(x).}$

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or, equivalently,

${\displaystyle h'=(f\circ g)'=(f'\circ g)\cdot g'.}$

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

${\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}$

and

${\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},}$

for indicating at which points the derivatives have to be evaluated.

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