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 $y=g(x)$ 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 $f'(y).g'(x)$, and substituting $y=g(x)$ we have the chain rule

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

In traditional "d" notation we write

${\frac {\mathrm {d} z}{\mathrm {d} x}}={\frac {\mathrm {d} z}{\mathrm {d} y}}\cdot {\frac {\mathrm {d} y}{\mathrm {d} x}}.\,$

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 $\mathrm {D} F$ at $a\in \mathbf {R} ^{n}$ and G having derivative $\mathrm {D} G$ at $F(a)\in \mathbf {R} ^{m}$. Thus $\mathrm {D} F$ is a linear map from $\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{m}$ and $\mathrm {D} G$ 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

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

See also

• Chain (mathematics)