So if I'm taking the derivative of the left-hand side, the same thing as taking the derivative of the right-hand side, the fundamental theorem of calculus tells us that this is going to be equal to f, lowercase f of x. Now, why is this a big deal? Why does it get such an important title as the fundamental theorem of calculus? Well, it tells us that for any continuous function f, if I define a function that is the area under the curve between a and x right over here, that the derivative of that function is going to be f. So let me make it clear. Every continuous function, every continuous f has an antiderivative capital F of x. That by itself is a cool thing. But the other really cool thing, or I guess these are somewhat related, is remember, coming into this, all we did, we just viewed the definite integral as symbolizing the area under the curve between two points.