# Chain rule

In calculus, the **chain rule** describes the derivative of a "function of a function": the composition of two function, where the output *z* is a given function of an intermediate variable *y* which is in turn a given function of the input variable *x*.

Suppose that *y* is given as a function and that *z* is given as a function . The rate at which *z* varies in terms of *y* is given by the derivative , and the rate at which *y* varies in terms of *x* is given by the derivative . So the rate at which *z* varies in terms of *x* is the product , and substituting we have the *chain rule*

In order to convert this to the traditional (Leibniz) notation, we notice

and

- .

In mnemonic form the latter expression is

which is easy to remember, because it as if d*y* in the numerator and the denominator of the right hand side cancels.

## Multivariable calculus

The extension of the chain rule to multivariable functions may be achieved by considering the derivative as a *linear approximation* to a differentiable function.

Now let and be functions with *F* having derivative at and *G* having derivative at . Thus is a linear map from and is a linear map from . Then is differentiable at with derivative