Question

What is the chain rule?

Answer 1

In calculus, the chain rule is a formulafor computing the derivative of the composition of two or more functions. That is, if f and g are functions, then the chain rule expresses the derivative of their composition f ∘ g (the function which maps x to f(g(x)) ) in terms of the derivatives of f and g and the product of functions as follows:

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

This may equivalently be expressed in terms of the variable. Let F = f ∘ g, or equivalently, F(x) = f(g(x)) for all x. Then one can also write

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

The chain rule may be written in Leibniz's notation in the following way. If a variable z depends on the variable y, which itself depends on the variable x, so that y and z are therefore dependent variables, then z, via the intermediate variable of y, depends on x as well. The chain rule then states,

{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}.}

The two versions of the chain rule are related; if {\displaystyle z=f(y)} and {\displaystyle y=g(x)}, then

{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=f'(y)g'(x)=f'(g(x))g'(x).}

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