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 \,y = g(x) and that z is given as a function \,z = f(y). The rate at which z varies in terms of y is given by the derivative \, f'(y), and the rate at which y varies in terms of x is given by the derivative \, g'(x). So the rate at which z varies in terms of x is the product \,f'(y)\sdot g'(x), and substituting \,y = g(x) we have the chain rule

(f \circ g)' = (f' \circ g) \sdot g' . \,

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

 z(y(x))\quad \Longleftrightarrow\quad  z\circ y(x)


 (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} . \, .

In mnemonic form the latter expression is

\frac{\mathrm{d} z}{\mathrm{d} x} = \frac{\mathrm{d} z}{\mathrm{d} y} \, \frac{\mathrm{d} y}{ \mathrm{d} x} , \,

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

[edit] 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 F : \mathbf{R}^n \rightarrow \mathbf{R}^m and G : \mathbf{R}^m \rightarrow \mathbf{R}^p be functions with F having derivative DF at a \in \mathbf{R}^n and G having derivative DG at F(a) \in \mathbf{R}^m. Thus DF is a linear map from \mathbf{R}^n \rightarrow \mathbf{R}^m and DG is a linear map from \mathbf{R}^m \rightarrow \mathbf{R}^p. Then F \circ G is differentiable at a \in \mathbf{R}^n with derivative

\mathrm{D}(F \circ G) = \mathrm{D}F \circ \mathrm{D}G . \,

[edit] See also

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