Time to see it. Pick a function, then reveal its derivative (slope) and integral (area). This is the same demo used in the real NMA course.

For each function, do this:
1. Look at the function alone first.
2. Predict in your head: what will the slope (derivative) look like? Where is it steep, flat, zero?
3. Then toggle it on and check if you were right.
4. Do the same for the integral (the area building up).

Why predict before you peek? Your brain locks in what it generates far better than what it just reads — psychologists call it the generation effect. The half-second of guessing wrong is exactly when the learning happens. Don't skip it.

Did you spot the weird one? The exponential function's derivative and integral look identical to the function itself. It's the rare shape whose rate-of-change equals its own height. That special property is why the exponential shows up everywhere in neuroscience — from a neuron's voltage decaying to populations growing.

That property has a name: the exponential is an _eigenfunction_.

Differentiate some functions and you get back the same function times a number. The exponential is the famous case — the shape never changes, it just gets scaled:

d/dt [ e^(a·t) ] = a · e^(a·t)

Differentiating e^(a·t) gives back e^(a·t) scaled by a. A function that survives an operation unchanged-except-for-scaling is an eigenfunction of that operation. (The same holds when you integrate it.)

This is the exact same idea as eigenvectors in linear algebra: a vector that a matrix only stretches (without rotating) is an eigenvector. Here, a function that differentiation only scales is an eigenfunction — same concept, one level up. Eigenfunctions are what make the hard math of neuron models solvable by hand.