Differentiation of integrals

Okay, imagine you have a big box of candy. You know how many pieces are in the box, but you don't know which flavors are in there. You want to figure out what flavors are inside the box.

To do this, you start taking one piece out at a time and tasting it. Each time you take a piece out, you write down what flavor it is. After you taste a lot of pieces, you have a pretty good idea of what flavors are in the box.

Now let's say you want to know how much candy you've eaten so far. You look at your list of flavors and count how many pieces you've eaten of each flavor. You add up all those numbers and you get the total amount of candy you've eaten.

This is kind of like differentiation and integration, but with math instead of candy. When you differentiate, you're "taking out" tiny little pieces of a function and figuring out what their "flavors" are. Those "flavors" are called the derivative of the function.

When you integrate, you're doing the opposite - you're adding up all those tiny little pieces to get the total "amount" of the function. That amount is called the integral of the function.

So basically, differentiation and integration are like taking apart and putting back together a function - just like taking apart and putting back together a box of candy!