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Fréchet derivative chain rule
Theorem
| $X,Y,Z$ … Banach spaces with topology |
| $F\in C(X,Y)$ |
| $G\in C(Y,Z)$ |
| $ 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: Chain rule, Chain rule (disambiguation)
Parents
Requirements
