Chain rule - Aetherwisp

The chain rule is a formula that expresses the derivative of the composition of two functions. Given two real univariate functions $f(t)$ and $g(x)$ , their composition is $h(t)=g(f(t))=(g\circ f)(t)$ . The derivative of $h$ is

h'(t)=g'(f(t))f'(t)

This can be extended to arbitrary dimensions. Say now that $f$ and $g$ are defined as $f:\mathbb{R}^{N}\to \mathbb{R}^{M}$ and $g:\mathbb{R}^{M}\to \mathbb{R}^{L}$ . The composition is now $h:\mathbb{R}^{N}\to \mathbb{R}^{L}$ . The Differential of $h$ is given by

Dh(t)=D(g\circ f)(t)=Dg(f(t))\cdot Df(t)

The dot operator " $\cdot$ " represents a product of linear operators. If they are given in matrix form (i.e. the Jacobians), it is a matrix multiplication.

For a partial derivative of the $a$ -th component of $h$ in the $b$ -th argument of $f$ , and calling $x_{j}$ the arguments of $g$ , it reads as

\frac{ \partial h_{a} }{ \partial t_{b} }(t)=\sum_{j}\left.{\frac{ \partial g_{a} }{ \partial x_{j} }}\right|_{x=f(t)} \frac{ \partial f_{j} }{ \partial t_{b} } (t)

where the vertical bar denotes that the derivative of $g_{a}$ is evaluated in $f(t)$ .