Chain rule

In calculus, the chain rule is a formula for determining the derivative of a composite function. The counterpart of the chain rule in integration is the substitution rule.

Given that two functions, f and g, are differentiable, the chain rule can be used to express the derivative of their composite, f ⚬ g, also written as f(g(x)). The chain rule is as follows:

(f ⚬ g)' = (f ' ⚬  g) · g'

Let F = f ⚬ g (F(x) = f(g(x)), then the chain rule can also be written in Lagrange's notation as:

F '(x) = f '[g(x)]g'(x)

The chain rule can also be written using Leibniz's notation given that a variable y depends on a variable u which is dependent on a variable x. In such a case, y also depends on x via the intermediate variable u:


See also derivatives, quotient rule, product rule.