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7-2-35 fundamental theorem of calculus
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    In this problem, we're going to use the fundamental theorem of calculus because we're finding the derivative of an integral. So what we're going to do when we find the derivative of an integral, we're going to have DDX of the one side and we're going to have DDX of the other side. Now we know by the fundamental theorem of calculus that an integral and a derivative basically undo themselves. However, we have functions on the top and on the bottom. So the left side is going to be turning into DYDX. On the right side, we're going to stick in the upper limit. So we have the natural log of sqrt X ^2, but then we need to multiply it by the derivative of the upper function, IE the concept of the chain rule. Here we're going to subtract, we're going to stick in the bottom function, and then we have to times it by the derivative of the bottom function because of the chain rule concept. So when we compute this, we can see that our derivative is going to equal the natural log of sqrt X ^2. That means that it has to be a positive. When we know that the derivative of a natural logarithm, now we know that the logarithm has to be positive. So that's our absolute value derivative of X ^2 is going to be two X I'm going to put that out in front. So two XLN absolute value of X -, sqrt X ^2 once again is going to be X. So we get the natural log of the absolute value of X over sqrt 21 and then the derivative of X ^2 / 21 is going to be two X / 21. We could actually simplify this even more. We could see that this is 2 XLN absolute value of X -, 2 X over 21 lane absolute value of X. And because this is subtraction, we could think of it as minus a negative and split those terms up to X / 21 natural log of sqrt 21. Now these two terms could combine. If I have 2X natural log of X -, 2 X over 21 natural log of X, that's going to give me 2020 ones times 2X natural log of the absolute value of X Plus this square root could actually come out in front as 1/2, so we would get 1/2 of two X / 21 natural log of 21. If we take this another step or two, the two and the 20 would give us 40 / 21 X natural log of the absolute value of X plus the half and the two in the numerator and denominator will cancel giving us X / 21 natural log of 21.