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Let's say that we have capital f(x) is being equal to the definite integral from pi to x of cot^2t dt.

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and what we're curious about finding is the derivative of this business

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we want to figure out what f prime of x is equal to

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well this is a direct application of the fundamental theorem of calculus

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this is going to be the derivative with respect to x of all of this craziness

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so let me just copy and paste that

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so copy and paste

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so this is going to be the derivative with respect to that

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and the fundamental theorem of calculus tells us that this is going to be equal to -this is going to be equal to just this function

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but it's not going to be a function of t anymore

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its going to be a function of x

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so it's going to be equal to cot^2x

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and you'll often see problems like this is if you go to calculus competitions or things like that or maybe certain exams

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and it's like "Oh my God I have to take the antiderivative of all of this business, evaluate it at the different boundaries

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and then I've got to take the derivative." No! You just have to apply the fundamental theorem of calculus.

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And it's actually very straightforward and a very fast thing to do.

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Now, let's mix it up a little bit.

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Let's say that you had the expression -the definite integral from pi -instead of from pi to x, let's say it's from pi to x^2.

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Let me write that x^2 in a different color just to make it clear what I'm doing.

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From pi to x^2 of cot^2t dt, and you wanted to take the derivative of this business.

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So you wanted to take the derivative with respect to x -you want to take the derivative with respect to x of this business.

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How would you do it? Well the recognition here is is that this is -capital f(x) was defined as this.

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Now instead of an x you have an x^2, so this is the exact same thing as taking the derivative with respect to x of F of not x

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that would be this, instead we have F(x^2) where we had an x here before we now have an x^2

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you can verify, if you had F(x^2) every place you saw the x here would be an x^2, so it would look exactly like this.

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And so we just have to apply the chain rule.

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So this is going to be equal to -and this is going to be equal to the derivative of F with respect to x^2

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so this is going to be -and this is straight out of the chain rule -it's going to be F prime of x^2 times the derivative of with respect to x of x^2.

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This is the derivative of F with respect to x^2

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and then times the derivative of x^2 with respect to x

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This is just the chain rule right over here

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So what's F prime of x^2 -the derivative with respect to x^2?

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Well, if you evaluate F prime at x^2, instead of just being cot^2x, it's going to be cot^2x^2.

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So this part right over here -this business right over here is going to be cot^2x^2, so this is the derivative of all this business with respect to x^2

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And then you have to multiply that times the derivative of x^2 with respect to x, which is just 2x

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So there you have it, the derivative of this, all this craziness, is equal to 2x cot^2x^2.