Finding derivative with fundamental theorem of calculus: chain rule

let's say that we have the function capital f of X which we're going to define is the definite integral from 1 to sine of X so that's an inner interesting upper bound right over there of 2t minus 1 and of course DT and what we are curious about is trying to figure out what is f prime of X going to be equal to so pause this video and see if you can figure that out alright so some of you might have been a little bit challenged by this notion of hey instead of an X on this upper bound I now have a sine of X if it was just an X I could have used the fundamental theorem of calculus just to review that if I had a function let me call it H of X if I have H of X that was defined as the definite integral from 1 to X of 2 t minus 1 DT we know from the fundamental theorem of calculus that H prime of X would be simply this inner function with the T replaced by the X it would just be 2x minus 1 pretty straightforward but this one isn't quite as straightforward stead of having an X up here our upper bound is a sine of X so one way to think about it is if we were to define G of X as being equal to sine of X is equal to sine of X our capital f of X can be expressed as capital f of X is the same thing as H of H of instead of an X everywhere we see an X we're replacing it with a sine of X so it's H of G of X G of X you can see the G of X right over there so you replace X with G of X for where in this expression you get H of G of X and that is capital f of X now why am i doing all of that well this might start think making you think about the chain rule because if this is true then that means that capital f prime of X is going to be equal to H prime of G of X H prime of G of X times G prime of X and so what would that be well we already know what H prime of X is so let me just in another color this part right over here is going to be equal to everywhere we see an X here we'll replace with the G of X so it's going to be 2 2 times sine of X 2 sine of X and then minus 1 minus 1 this is this right over here and then what's G prime of X G prime of X well G prime of X is just of course the derivative of sine of X is cosine of X is cosine of X so this part right over here is going to be cosine of X and we could keep going we could try to we could try to simplify this a little bit or rewrite it in different ways but there you have it