Chain rule using the Jacobian for general spaces If $f:X \rightarrow Y$ is differentiate at $x$ and $g:Y \rightarrow Z$ is differentiable at $y = f(x)$, then $g \circ f: X \rightarrow Z$ is differentiable at x and $$F(g\circ f)(x) = Dg(f(x)) Df(x)$$
I am able to prove it for univariate functions or for $\mathbb{R}^n \rightarrow \mathbb{R}$ functions, but not for more general spaces such as $X, Y$ And $Z$. Can somebody help me on that?

I like the Weierstraß formulation of differentiation for such cases:
$f$ is differentiable at $x$ if there is a linear function $L$ and a remainder function $r$ such that $f(x+v)=f(x)+L(v)+r(v)$ with $\lim_{v\to 0}\frac{r(v)}{\|v\|}=0.$
This has the advantage that you do not have to bother coordinates or specific directions. This way we get matrix multiplication for the derivatives and only have to work on the remainders a bit.
