# Difference between revisions of "Chain rule"

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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 dy 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