===== Differentiable function =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $X,Y$ ... Banach spaces with topology |
| @#55CCEE: context     | @#55CCEE: $n\in \mathbb N$ |
| @#FFBB00: definiendum | @#FFBB00: $f\in C^n(X,Y)$ |
| @#55EE55: postulate   | @#55EE55: $\forall(k\le n).\,D^k f$ ... 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: [[http://en.wikipedia.org/wiki/Differentiable_function|Differentiable function]]

-----
=== Context ===
[[Fréchet derivative]], [[Iterated function]]