The Derivative Operator

In standard calculus, the derivative is written $\frac{d}{dx}f(x)$ — an expression that involves a specific variable name. This is ambiguous in complex situations. Is $x$ a free variable? Which $x$?

Sussman and Wisdom replace expression derivatives with functional derivatives. The operator $D$ takes a function and returns a new function — its derivative:

$(Df)(x) = \lim_{h \to 0} \frac{f(x+h) - f(x-h)}{2h}$

This is unambiguous: $D$ is a higher-order function (a function that takes and returns functions).

Examples

$D(x \mapsto x^2) = (x \mapsto 2x)$ $D(x \mapsto \sin x) = (x \mapsto \cos x)$ $D(D(x \mapsto x^3)) = (x \mapsto 6x)$

Numerical Implementation

We approximate $D$ using the central difference formula:

$Df(x) \approx \frac{f(x+h) - f(x-h)}{2h}$

With $h = 10^{-7}$, this is accurate to about 14 significant figures for smooth functions.

Your Task

Implement D(f) that returns the numerical derivative of f as a new function.