How come reducing the power of a function by 1 and multiplying the function by it gives us derivative of the function?

Question

Is it just a coincidence, or is it like an assumption and everything else is built around it?

In: 0

Answer 1

No, it is not a coincidence and not an assumption. It is a consequence of a fact, that “x^(n)” is a “volume” of an n-dimensional “cube”.

Derivative answers the question: how much does a function grow, if we make “x” slightly larger? In case of “x^(n)”, it means: how much does a volume of nD-cube grows, when we make side length slightly larger?

* The nD-cube has “n” directions to grow
* The “grow border” in each direction is an (n-1)-dimensional cube.

So, the total growing rate is equal (number of directions)*(border size in each direction) = n*x^(n-1)