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Mathematics-Online lexicon:

Multivariate Chain Rule

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

Let $ f$ be the composition

$\displaystyle h = g \circ f : x \mapsto y =f(x) \mapsto z = g(y)\, ,

of continuous differentiable functions $ f:\mathbb{R}^m \to \mathbb{R}^\ell$ and $ g:\mathbb{R}^\ell\to \mathbb{R}^n .$


$\displaystyle h^\prime(x) = g^\prime(y) f^\prime(x)\, ,

i.e. the jacobian matrix of $ h$ ist the product of the jacobian matrices of $ f$ and $ g$. The entries of $ h^{\prime}$ result from matrix multiplication.

$\displaystyle \frac{\partial h_i}{\partial x_k} = \sum_j \frac{\partial
g_i}{\partial y_j} \frac{\partial f_j}{\partial x_k}\, .

In particular in the special case $ m=n=1 $ (i.e. $ f$ is a parametric curve and $ g$ a scalar function of $ l$ variables) the chain rule has the form

$\displaystyle \frac{d\, h}{d \,x} = \left(\operatorname{grad} g\right)^{\operatorname{t}} f^{\prime}(x)


[Examples] [Links]

  automatically generated 8/ 4/2008