## 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

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

Wikipedia: Chain rule, Chain rule (disambiguation)