 Hi, and welcome to another example of using the chain rule. In this video, I'd like to go back and find the derivative of this function. C of x equals the square root of x cubed minus two. And I mentioned that function in the review video for this section. So let's find c prime of x. Now this isn't as straightforward as it sounds because we have a lot of rules in front of us for taking derivatives. And we need to be careful to select the right approach. So let's first of all think about why the chain rule would even apply here. Now the chain rule is the rule to use here as opposed to the product rule or the quotient rule or anything else because c of x is a composite function. Now how do I know it's a composite function, you ask? Well, hopefully we're getting good at spotting composite functions at this point but if not, here's an easy way to check. Just pick a value of x in the domain of the function, like in this case x equals three. And just evaluate the function at this point and pay attention to the processes we use. In this case I have to substitute x equals three and then I have to go through two stages. I find three cubed minus two first. Then I take the result of that and find the square root of it. So the fact that finding c of three took two distinct stages. I'm evaluating at the x cubed minus two function. And then running the output of that through the square root function means I have two functions that are linked together. That's a composite function. So this function again is really two functions, x cubed minus two on the one hand and then square root of x. They're chained together, so we do one and then we do the other. Since c is a composite to differentiate it, I must use the chain rule. So here's a restatement of the definition of the chain rule. And I have a concept check for you. For this particular function, what function plays the role of f and what function plays the role of g in the chain rule formula? So pause the video and think about this. Look over the options here. Select the one that works the best and then come back with your selection. So the correct answer here is b. And for the reason, just think back to the little thought experiment I used for determining how c of x was composite in the first place. That was by evaluating c of three. When I did that, I had to compute two functions in a particular order. I first had to find three cubed minus two, that was the first function. And then I took the square root of the result and the square root was the second function. This tells me that the quote unquote inside function, which I'm calling g of x in the definition of the chain rule. That's the first link in the chain is the function x cubed minus two. And the outside function, or the second and final link in the chain, is square root of x. Choice a is not correct because this selection of f and g gives the wrong function when you compose them. In this case, f of g of x would be the function x cubed minus two, with square root of x plugged in. And that would give me x to the three halves minus two. And that's not my original function c. So choice a is not correct and choice e is not correct either. Choices c and d are not totally right. But if you selected one of them, you're not totally wrong either. We'll come back to that in just a minute. For now, let's go over to my handwriting slide and let's find the derivative of c of x using the chain rule. So here we are, we're going to take the derivative of radical x cubed minus two. And we've identified that this is a composite function, and that's the first and most important step in performing this derivative calculation because that's going to tell us what rule to use. It's a composite function, and in the language of the chain rule, we decided the g of x, which is the inside function or first link in the chain was x cubed minus two. And the outside function f of x, which is the second link and final link in the chain is radical x. And it's going to be helpful to rewrite this as x to the one half power because we have some derivatives coming up. So in using the chain rule, we are going to just first of all write down the definition of the chain rule, which says that c prime of x would be f prime of g of x times the derivative of g. So now it's just really a matter of computing all the ingredients to this formula here that I've written down and putting them in the right place. So let's do that off to the side. Here is g of x, and so g prime of x is quite simple. That's just three x squared, and f of x is x to the one half. So therefore f prime of x is one half x to the minus one half. So now we're just simply going to assemble all the pieces according to what the chain rule tells us. Now what it says is I have to take f prime, which is this over here, and put not x into it, but g of x into it. So I'm going to write f prime, but leaves sort of a blank here for the input. There is the basic formula shape for f prime of x. And what I'm putting into this is not x, but it's g of x. It's the original g of x I have right here. So let me actually keep my green and put x cubed minus two right into that. That is f prime of g of x. Secondly, I would need to multiply this by, that's a dot there for multiplication, times g prime of x. And that's right over here, that's three x squared. And that's the basic derivative here. I'm just going to do one cosmetic algebra step here, not really to simplify, but just to make it look a little bit nicer. And that is I have a one half and a three and an x squared. I'm going to write this as three x squared divided by two radical x cubed minus two. That's just rearranging some of the pieces here. But the real derivative is done right there. And again, what's important about this is, first of all, noticing that c is a composite function in the first place, and then pulling it apart in the right way. So we'll know what the inside function is and what the outside function is. So I said that earlier, if you had selected c or d in this concept check, you wouldn't have been totally wrong. Now what I mean by that is that for both choices of f and g in these two options, you do actually get a correct decomposition of the function c. And choice c, for example, f of g of x, if you compose them, would be square root of x minus two with x cubed plugged in for x. And that does actually give you the square root of x cubed minus two. And this also works for choice d. So there's more than one way to decompose a composite function. However, when we think about the chain rule, notice that eventually I will need to take two derivatives, the derivative of f, the outside function, and the derivative of g, the inside function. If I chose f and g so that it's not easy to differentiate one of these, then I have messed myself up. For example, in c, when I go to differentiate square root of x minus two, well, how would I do that? This is another composite function, and I would have to use the chain rule a second time. And that's not very easy. Likewise, in d, if I chose this selection of f and g, I would need to take the derivative of square root of x cubed minus two. But if I knew how to do that, then I wouldn't be doing this problem in the first place. So although a, I'm sorry, although b and c and d all give correct decompositions of my function, only b is the best choice for doing the chain rule. So we're gonna choose f and g in the chain rule, in other words, so that not only is the composite correct, but also the derivatives involved are easy. Thanks for watching.