Differentiating using multiple rules: strategy | AP Calculus AB | Khan Academy

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 is to look at the big picture structure of the expression. One way to think about it is 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 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 into 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 (CR 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.

And 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—I'm just going to make that pink circle for the something rather than writing it all again—of s of that something, sine of that something, not even thinking about what that something is just yet, times the derivative with respect to x of that something.

This is just an application of the chain rule. No matter what was 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 sine, I would move to this step.

The derivative with respect to that something of sine of that something times the derivative with respect to x of that 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 x² + 5 times cosine of x. That's that circle right over there. Let me 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 x² + 5 times cosine of x.

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 input into like an s 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. I could keep doing that and compute it, and 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 a 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 says, “Okay, I 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, then I would use the quotient rule. But 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'm going to 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 s of x² + 5 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 to just show how you identify the structures in these expressions to think about, well, 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.