Functional differentiation
1 Introduction
Functional differentiation means computing or approximating the derivative of a function. There are several ways to do this:
- Approximate the derivative f'(x) by
where h is close to zero. (or at best the square root of the machine precision 
.
- Compute the derivative of f symbolically. This approach is particularly interesting for Haskell.
2 Functional analysis
If you want to explain the terms Higher order function and Currying to mathematicians, this is certainly a good example. The mathematician writes
and the Haskell programmer writes
derive :: a -> (a -> a) -> (a -> a) derive h f x = (f (x+h) - f x) / h .
Haskell's derive h approximates the mathematician's D.
In functional analysis D is called a (linear) function operator, because it maps functions to functions.
In Haskell derive h is called a higher order function for the same reason.
D is in curried form. If it would be uncurried, you would write D(f,x).
3 Blog Posts
There have been several blog posts on this recently. I think we should gather the information together and make a nice wiki article on it here. For now, here are links to articles on the topic.