Differentiable function

Set

context $X,Y$ … Banach spaces with topology
context $n\in \mathbb N$
definiendum $f\in C^n(X,Y)$
postulate $\forall(k\le n).\,D^k f$ … continuous

Theorems

Let

$f(0)=0\neq f'(0)$,

then

$\dfrac{ f(y\ f^{-1}(x)) }{y} =x+(y-1)\cdot\dfrac{f''(0)}{f'(0)^2}\cdot\dfrac{1}{2}x^2+{\mathcal O}(x^3)$

Reference

Wikipedia: Differentiable function


Context

Fréchet derivative, Iterated function