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# Differentiating products

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.B (LO)
,
FUN‑3.B.1 (EK)

## Video transcript

so let's see if we can find the derivative with respect to X F e to the x times cosine of X and like always pause this video and give it a go on your own before we work through it so when you look at this you might say well I know how to find the derivative with respect to e to the X that's in fact just e to the X and let me write this down we know we know a few things we know the derivative with respect to X of e to the X e to the X is e to the X we know we know how to find the derivative of cosine of X the derivative with respect to X of cosine of X is equal to negative sine of X but how do we find the derivative of their product well as you can imagine this might involve the product rule and let me just write down the product rule generally first so if we take the derivative with respect to X of the first expression in terms of X so this is we could call this U of x times another expression that involves X so U times V of X this is going to be equal to and I'm color coding it so we can really keep track of things this is going to be equal to the derivative of the first expression so I could write that as u prime of x times just the second expression not the derivative of just the second expression so times V of X and then we have plus plus the first expression not its derivative just the first expression U of x times the derivative of the second expression times the derivative of the second expression so what you remember it is you have to these two things here is you're going to end up with two different terms and each of them you're going to take the derivative of one of them but not the other one and then the other one you'll take the derivative of the other one but not the first one so u derivative of U times V is a u prime times V Plus u times V Prime now when you just look at it like that it seems a little bit abstract and that might even be a little confusing but that's why we have a tangible example here and I color-coded it intentionally so we can say that U of X is equal to e to the X and V of X is equal to cosine of X so V of X is equal to cosine of X and if u of X is equal to e to the X we know that the derivative of that with respect to X is still e to the X that's one of the most magical things in mathematics one of the one of the things that makes ysou special so u prime of X is still equal to e to the X and V prime of X V prime of X we know is negative sine of X negative sine of X and so what's this going to be equal to this is going to be equal to the derivative of the first expression so the derivative of e to the X which is just e to the x times the second expression not taking its derivative so times cosine of X plus plus the first expression not taking its derivative so e to the x times the derivative of the second expression so times the derivative of cosine of X which is negative sine negative sine of X and it might be a little bit confusing because e to the X is its own derivative but this right over here you can view this is this was the derivative of e to the X which happens to be e to the X that's what's exciting about that that expression or that function and then this is just easy next without taking straight out there of course the same thing but anyway well now we can just simplify it this is going to be equal to this is going to be equal to we could write this either as e to the x times cosine of X times cosine of X minus e to the X e to the X times sine of X times the sine of X or if you want you could factor out an e to the X this is the same thing as e to the x times cosine of X minus sine of X cosine of X minus sine of X so hopefully this makes the the product rule a little bit more tangible and once you have this in your tool belt there's a whole broader class of functions and expressions that we can start to differentiate
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