# 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 be equipped with the derivative : .
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:

- fad
- Vector-space
- http://comonad.com/haskell/monoids/dist/doc/html/monoids/Data-Ring-Module-AutomaticDifferentiation.html

## Power Series

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

Implementation with Haskell 98 type classes: http://darcs.haskell.org/htam/src/PowerSeries/Taylor.hs

With advanced type classes in Numeric Prelude: http://hackage.haskell.org/packages/archive/numeric-prelude/0.0.5/doc/html/MathObj-PowerSeries.html

## See also

- Functional differentiation
- Chris Smith in Haskell-cafe on Hit a wall with the type system