Chain rule: Difference between revisions

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In [[calculus]], the '''chain rule''' describes the [[derivative]] of a "function of a function": the [[composition (mathematics)|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''.
In [[calculus]], the '''chain rule''' describes the [[derivative]] of a "function of a function": the [[composition (mathematics)|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 <math>y = g(x)</math> and that ''z'' is given as a function <math>z = f(y)</math>.  The rate at which ''z'' varies in terms of ''y'' is given by the derivative <math>f'(y)</math>, and the rate at which ''y'' varies in terms of ''x'' is given by the derivative <math>g'(x)</math>.  So the rate at which ''z'' varies in terms of ''x'' is the product <math>f'(y).g'(x)</math>, and substituting <math>y = g(x)</math> we have the ''chain rule''
Suppose that ''y'' is given as a function <math>\,y = g(x)</math> and that ''z'' is given as a function <math>\,z = f(y)</math>.  The rate at which ''z'' varies in terms of ''y'' is given by the derivative <math>\, f'(y)</math>, and the rate at which ''y'' varies in terms of ''x'' is given by the derivative <math>\, g'(x)</math>.  So the rate at which ''z'' varies in terms of ''x'' is the product <math>\,f'(y)\sdot g'(x)</math>, and substituting <math>\,y = g(x)</math> we have the ''chain rule''
: <math>(f \circ g)' = (f' \circ g) \sdot g' . \,</math>


: <math>(f \circ g)' = (f' \circ g) . g' . \,</math>
In order to convert this to the traditional ([[Leibniz]]) notation, we notice
:<math> z(y(x))\quad \Longleftrightarrow\quad  z\circ y(x) </math>
and
:<math> (z \circ y)' = (z' \circ y) \sdot y' \quad \Longleftrightarrow\quad
\frac{\mathrm{d} z(y(x))}{\mathrm{d} x} = \frac{\mathrm{d} z(y)}{\mathrm{d} y} \, \frac{\mathrm{d} y(x)}{ \mathrm{d} x} . \, </math>.
In mnemonic form the latter expression is
:<math>\frac{\mathrm{d} z}{\mathrm{d} x} = \frac{\mathrm{d} z}{\mathrm{d} y} \, \frac{\mathrm{d} y}{ \mathrm{d} x} , \, </math>
which is easy to remember, because it as if d''y'' in the numerator and the denominator of the right hand side cancels.


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. 


:<math>\frac{\mathrm{d} z}{\mathrm{d} x} = \frac{\mathrm{d} z}{\mathrm{d} y} \cdot \frac{\mathrm{d} y}{ \mathrm{d} x} . \, </math>
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
 
:<math>\mathrm{D}(F \circ G) = \mathrm{D}F \circ \mathrm{D}G . \,</math>


==See also==
==See also==
* [[Chain (mathematics)]]
* [[Chain (mathematics)]]

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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

See also