Fréchet derivative chain rule

Theorem

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.

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

Parents

Context

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