# Functional differentiation

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## 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 $\frac{f(x+h)-f(x)}{h}$ where h is close to zero. (or at best the square root of the machine precision $\varepsilon$.
• 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

$D f (x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$

derive :: a -> (a -> a) -> (a -> a)
derive h f x = (f (x+h) - f x) / h    .
derive h
approximates the mathematician's D.

In functional analysis D is called a (linear) function operator, because it maps functions to functions.

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.