5.3 The Fundamental Theorem of Calculus [STARTS AT 11:47] INSTRUCTOR: A little different in our limits. We've got x to 2x. The way we're going to do this is let's start with F of x. Let's split this into the integral from x to 0 t cubed dt plus the integral from 0 to 2x. We can split it by any value we would like. And just so happens that 0 is a convenient splitting value. Now, we're going to switch because we want the x limit as the upper limit. So this will be negative 0 to x at integral dt plus integral from 0 to 2x t cubed dt. Now, taking the derivative of both sides-- I'm just going to mark this up. So the derivative with respect to x. We are going to have negative x cubed. All right. And that will be plus x cubed times the derivative of 2x, which is 2. Oh actually, let's switch something there. That is going to be 2x cubed times 2. There we go. Because of our chain rule, which will then be equal to 16x cubed. So our F prime of x is equal to 15x cubed. All right. [ENDS AT 13:51]