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# Differentiating using multiple rules: strategy

AP.CALC:
FUN‑3 (EU)

## Video transcript

so I have two different expressions here that I want to take the derivative of and what I want you to do is pause the video and think about how you would first approach taking the derivative of this expression and how that might be the same or different as your approach in taking the derivative of this expression the goal here isn't to compute the derivatives all the way but really to just think about how we identify what strategies to use okay so let's first tackle this one and the key when looking at a complex expression like either of these it's look at the big picture structure of the expression so one way to think about it let's look at the outside rather than the inside details so if we look at the outside here we have the sine of something so there's a sine of something going on here that I'm going to circle in red or in this pink color so that's not how my brain thinks about it from the outside I'm like okay big picture I'm taking the sine of some stuff I might be taking some stuff to some exponent in this case I'm inputting it a trigonometric expression but if you have a situation like that it's a good sign that the chain rule is in order so let me write that down so we would want to use in this case the chain rule see our for chain rule and how would we apply it well we would take the derivative of the outside with respect to this inside times the derivative of this inside with respect to X I'm going to write it the way that my brain sometimes thinks about it so we can write this as the derivative with respect to that something so just going to make that that pink circle for the something rather than writing it all again of sine of that something sine of that something not even thinking about what that something is just yet x times the derivative with respect to X of that something this is just an application of the chain rule no matter what here in this pink colored circle it might have been something with square roots and logarithms and whatever else as long as it's being contained within the sign I would move to this step the derivative with respect to that something of sign of that something times the derivative with respect to X of the something now what would that be tangibly in this case well this first part I will do it in orange this first part would just be cosine of x squared plus five times cosine of X so that's that circle right over there we close the cosine right over there and then times the derivative with respect to x times the derivative with respect to x of all of this again of x squared plus five times cosine of X and then I would close my brackets and of course I wouldn't be done yet I have more derivative taking to do here now I would look at the big structure of what's going on and I have two expressions being multiplied I don't have just one big expression that's being an input into like a sine function or cosine function or one big expression that's taken to some exponent I have two expressions being multiplied I have this being multiplied by this and so if I'm just multiplying two expressions that's a pretty good clue that to compute this part I would then use the product rule and I could keep doing that and compute it I encourage you to do so but this is more about the strategies and how do you recognize them but now let's go to the other example well this looks a lot more like this step of the first problem than the beginning of the original problem here I don't have a sine of a bunch of stuff or bunch of stuff being raised to one exponent here I have the product of two expressions just like we saw over here we have this expression being multiplied by this expression so my brain just have two expressions then I'm going to use the product rule two expressions being multiplied I'm going to use the product rule if it was one expression being divided by another expression that I would use the quotient rule in this case it's going to be the product rule and so that tells me that this is going to be the derivative with respect to x of the first expression just going to do that with the orange circle times the second expression I would do that with the blue circle plus the first expression not taking its derivative the first expression times the derivative with respect to X of the second expression once again here this is just the product rule you can substitute sine of x squared plus five where you see this orange circle and you can substitute cosine of X where you see this blue circle but the whole point here isn't to actually solve this or compute this but really they just show how you identify the structures in these expressions to think about what do I use the chain rule first and then use the product rule here or in this case do I use the product rule first and even once you do this you're not going to be done then to compute this derivative you're going to have to use the chain rule and you'll keep going until you don't have any more derivatives to take
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