I dont understand why integrals and derivatives work, and i dont understand why theyre so closely related. Let's take a function y= x^2 + 2x + 9 y' = 2x + 2 Why do the rules for taking derivatives work? Why does reducing the power of a term by 1 and adding that as a coefficient work to find the rate of change? This might be like asking why gravity exists, but i'm curious. Now in regards to my main question, take that derivative, y' = 2x + 2 and graph it. If you want to calculate the area of the curve under the derivative, you can just do so using the original term it's derived from (by dropping any constants that would get lost in the derivation). So i dont understand why a function can also serve to express the area under the function of it's derivative (when you drop constant terms. I know mathematically it makes sense because the process of derivation and antiderivation are opposites, so i mathematically understand the connection, but i dont get the relationship between RoC and area.