Fréchet derivative chain rule

context	$X,Y,Z$ … Banach spaces with topology
context	$f\in C(X,Y)$
context	$g\in C(Y,Z)$

postulate

$D(g\circ f)=(Dg)\circ f\ \cdot\ Df$

where $\circ$ denotes the concatenation of functions of $X,Y$ , which is taken to bind stronger than the concatenation $\cdot$ of linear operators.

For functions in $f,g: \mathbb R\to\mathbb R$ , this of course reads

$\frac{\partial}{\partial x}g(f(x))=g'(f(x))\cdot f'(x)$