# Automatic Differentiation

Automatic Differentiation roughly means that a numerical value is equipped with a derivative part, which is updated accordingly on every function application. Let the number x0 be equipped with the derivative x1: $\langle x_0,x_1 \rangle$. For example the sinus is defined as:

• \sin\langle x_0,x_1 \rangle = \langle \sin x_0, x_1\cdot\cos x_0\rangle

You see, that's just estimating errors as in physics. However, it becomes more interesting for vector functions.

Implementations:

## 1 Power Series

You may count arithmetic with power series also as Automatic Differentiation, since this means just working with all derivatives simultaneously.