===== Fréchet derivative chain rule =====
==== Theorem ====
| @#55CCEE: context     | @#55CCEE: $X,Y,Z$ ... Banach spaces with topology |
| @#55CCEE: context     | @#55CCEE: $f\in C(X,Y)$ |
| @#55CCEE: context     | @#55CCEE: $g\in C(Y,Z)$ |

| @#55EE55: postulate   | @#55EE55: $ 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.

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

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

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Chain_rule|Chain rule]], [[http://en.wikipedia.org/wiki/Chain_rule_%28disambiguation%29|Chain rule (disambiguation)]]
==== Parents ====
=== Context ===
[[Fréchet derivative]]