That's a sort of function, yes, but it's numerical analysis. In a Lisp you can pass in the equivalent of "4x^3 + (x^2)/2 -5X + 6" and get back "12x^2 + x - 5" - a function that is the derivative of the function you put in, rather than a function that approximates the value of the derivative at a value. You'd be manipulating the function rather than its return value.

Lisp is definitely not doing symbolic algebra and finding a pure analytical answer like that under the hood - symbolic math, differentiation rules etc. are a whole other beast (see SymPy for example).

Maybe that's the misunderstanding in the original article that made it seem so mindblowing?

Evaluating (f(5.0001) - f(5.0)) / 0.0001 just gives an approximation of the derivative at a certain point in the function (aka the "finite difference method").

Symbolic derivatives are one of the very next sections in the famous lisp lectures that this was watching.

Interesting - for symbolic differentiation I imagine it's about code-manipulation rules for e.g. calculus' product rule, chain rule etc.?

Essentially, yes. And if you want it to be pretty, it gets complicated quickly.

https://mitpress.mit.edu/sites/default/files/sicp/full-text/... is the chapter. Turns out it wasn't the "very next" one. But it is in my head.

Here's symbolic differentiation done in ~20 lines of Haskell: https://www.quora.com/What-are-20-lines-of-code-that-capture...