# Difference between revisions of "Chain rule"

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:<math>\frac{\mathrm{d} z}{\mathrm{d} x} = \frac{\mathrm{d} z}{\mathrm{d} y} \cdot \frac{\mathrm{d} y}{ \mathrm{d} x} . \, </math> | :<math>\frac{\mathrm{d} z}{\mathrm{d} x} = \frac{\mathrm{d} z}{\mathrm{d} y} \cdot \frac{\mathrm{d} y}{ \mathrm{d} x} . \, </math> | ||

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+ | ==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 <math>F : \mathbf{R}^n \rightarrow \mathbf{R}^m</math> and <math>G : \mathbf{R}^m \rightarrow \mathbf{R}^p</math> be functions with ''F'' having derivative <math>\mathrm{D}F</math> at <math>a \in \mathbf{R}^n</math> and ''G'' having derivative <math>\mathrm{D}G</math> at <math>F(a) \in \mathbf{R}^m</math>. Thus <math>\mathrm{D}F</math> is a linear map from <math>\mathbf{R}^n \rightarrow \mathbf{R}^m</math> and <math>\mathrm{D}G</math> is a linear map from <math>\mathbf{R}^m \rightarrow \mathbf{R}^p</math>. Then <math>F \circ G</math> is differentiable at <math>a \in \mathbf{R}^n</math> with derivative | ||

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+ | :<math>\mathrm{D}(F \circ G) = \mathrm{D}F \circ \mathrm{D}G . \,</math> | ||

==See also== | ==See also== | ||

* [[Chain (mathematics)]] | * [[Chain (mathematics)]] |

## Revision as of 12:11, 8 November 2008

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 traditional "d" notation we write

## 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