= Multivariable chain rule
{title2=$J_{g\circ f}(x)=J_g(f(x))J_f(x)$}
= Jacobian chain rule
{c}
{synonym}
For <differentiable maps> $f:\mathbb R^n\to\mathbb R^m$ and $g:\mathbb R^m\to\mathbb R^r$, the <multivariable chain rule> multiplies their <Jacobian matrices>:
$$
J_{g\circ f}(x)=J_g(f(x))J_f(x).
$$
The outer $r\times m$ <matrix> multiplies the inner $m\times n$ <matrix>, producing an $r\times n$ <matrix>. This order records that the inner <linear map> acts first. Componentwise, the <matrix multiplication> sums the contributions of all intermediate coordinates. Existence of separate <partial derivatives> alone is not the required <differentiability> hypothesis.
Back to article page
