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# Applying the chain rule and product rule

AP.CALC:
FUN‑3 (EU)

## Video transcript

what we're going to do in this video is try to find the derivative with respect to X of x squared sine of X all of that to the third power and what's going to be interesting is that there's multiple ways to tackle it and I encourage you to pause the video and see if you can work through it on your own so there's actually multiple techniques one path is to do the chain rule first so I'll just say cr4 chain rule first and so I have I'm taking the derivative with respect to X of something to the third power so if I take the derivative it would be the derivative with respect to that something so that would be three times that something squared times the derivative with respect to X of that something where the something in this case is x squared sine of X x squared sine of X this is just an application of the chain rule now the second part what would that be the second part here is in another color in orange well here I would apply the product rule I have the product of two expressions so I would take the derivative of let me write this down so this is going to be the product rule product rule I would take the derivative of the first expression so X derivative of x squared is 2x right a little bit to the right this is going to be 2x times the second expression sine of X plus the first expression x squared times the derivative of the second one cosine of X that's just the product rule as applied to this part right over here and all of that of course is being multiplied by this up front which actually let me just rewrite that so all of this I could rewrite as let's see this would be three times if I have the product things raised to the second power I could take each of them to the second power and then take the product so x squared squared is X to the fourth and then sine of x squared is sine squared of X and then all of that is being multiplied by that and if we want we can algebraically simplify we can distribute everything out in which case what would we get well let's see three times two is six x to the fourth times X is X to the fifth sine squared of x times sine of X is sine of X to the third power and then let's see three so plus three X to the fourth times x squared is X to the sixth power and then I have sine squared of X sine squared of X cosine of X so there you have it that's one strives you chain rule first and then product rule what would be another strategy pause the video and try to think of it well we could just algebraically use or expel exponent properties first in which case this is going to be equal to the derivative with respect to X of if I'm taking x squared times sine of X to the third power instead I could say X to the third to the third power which is going to be X to the sixth and then sine of X to the third power sine of X to the third power I'm using the same exponent property that we have we used right over here to simplify this if I have something the product things to some exponent well that's the same thing of each of them raised to the exponent and then the product of the two now how would we tackle this well I here I would do the product rule first so let's do that so let's do the product rule so we're going to take the derivative of the first expression so derivative x to the sixth is 6x to the fifth times the second expression sine to the third of X or sine of X to the third power plus the first X to the sixth times the derivative of the second and I'm just going to write that d DX of sine of X to the third power now to evaluate this right over here it does definitely make sense to use the chain rule we'll use the chain rule and so what is this going to be well I have the derivative of something to the third power so that's going to be three times that something squared times the derivative of that something so in this case if something is sine of X and the derivative of sine of X is cosine of X then I have all of this business over here I have 6x to the fifth sine to the third or sine of X to the third power plus X to the sixth and if I were to just simplify this a little bit in fact you see it very clearly these two things are equivalent this this term is exactly equivalent to this term the way it's written and then this is exactly if you multiply three times X to the sixth sine of x squared cosine of X so the nice thing about math if we're doing things that make logical sense we should get to the same end point but the point here is that there's multiple strategies you could use the chain rule first and then the product rule or you could use a product rule first and then the chain rule in this case you could debate which one is faster it looks like the one on the right might be a little bit faster but sometimes but there these two are pretty close but sometimes it'll be more clear than not which one is preferable you really want to minimize the amount of hairiness the amount number of steps the chances for careless mistakes you might have