eli5: WHY do derivatives and integrals work?

Question

I’m embarrassed to admit I’m getting my masters degree in a math related subject and I still don’t get this!

I know how to do them, but the way you compute them is almost suspiciously simple. What’s the logic behind converting the exponent to a constant? How does that determine the slope?

In: 0

Answer 1

It is a simple idea. For derivatives, you estimate the slope with two points, then take the limit as those two points get closer together. For why the power rule is the way it is:

let f(x) = x^n
f'(x) = lim[d->0] (f(x + d) – f(x)) / d
= lim[d->0] ((x+d)^n – x^(n)) / d
= lim[d->0] (x^n + nx^(n-1)d^1 + [a bunch of terms with d^2 or higher] – x^(n)) / d
= lim[d->0] (nx^(n-1)d^1 + [a bunch of terms with d^2 or higher]) / d
= lim[d->0] nx^(n-1) + [a bunch of terms with d^1 or higher]
= nx^(n-1)

For (Riemann) integrals, you estimate the area with a bunch of rectangles, then take the limit as the width of the rectangles goes to 0. Usually a little hairier, but the same idea.

Calculus is just geometry + limits.