# Functional differentiation

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## 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.

## 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

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

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 $D(f,x)$.

## 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.