How come the inverse of a derivative is an integral?

Question

I can’t conceptually grasp this the same why I can grasp addition/subtraction (the opposite of adding stuff is to take it away), but if the derivative of a function is the slope at any given point, then why would the area under the curve be the opposite of that?

In: 2

Answer 1

From my other answer to same question:

Slope indicates how rapidly the function changes its value as it goes on.

Derivative of the integral (i.e area) describes how rapidly the area grows, i.e. the initial function itself (since the larger the function value is, the more it adds to the area).

Integral of the derivative means adding all those little slopes together. At every point the slope points to where the function is going next, so integrating them will, again, trace the initial function.