 Let's take a look at an example of how we can use the chain rule and a table of values that we're given to find requested derivatives. So if you take a look at this table, we have two different x values, two and five, and it seems like we're dealing with two functions, f and g. We have the function values for those two functions as well as the derivatives. Now this last one didn't show up over here. I just noticed this should be a square root of two right there, just so you know. So what we're going to be asked in each of the following problems, there's three problems we're going to look at, is to find h prime of two specifically, okay? So keep that in mind. So the first one we're asked to do is when the function h is defined to be the composite function, f composed g, f of g of x. Now because it's a composite function, if we wanted to find the derivative of h, that's going to require us to do the chain rule. So remember how the chain rule goes. We start on the outside and we work our way in taking one derivative at a time. So the outside function in this case is f. So we're going to start with f prime and we're going to keep the g function as it is, remember. And now we have to go inside. We're going to have to take the derivative of the g function and just multiply by g prime of x. Now remember we specifically wanted h prime of two to be found, all right? So let's just make this easy. Fill in our twos for x's. And then we should be able to use the table of values to finish this off. All right, so let's start with this inside right here. g of two. So if you take a look at the table, g of two is five. So that means we have f prime of five. Before I forget, don't forget this value over here for some reason. That's the one that didn't show up, that's square root of two, just in case we need it. And actually we need it now because g prime of two is going to be that square root of two that we just filled in. Now right here, f prime of five, so that's pi according to the table. So we have pi times square root of two as our answer for this. So notice what it really requires you to understand is how to use the chain rule. When you don't have a specific function like a polynomial function or trig function, you have a very generally expressed function and you really need to understand how the chain rule goes. So let's look at another example. In this case, the function h is now still a composite function, but the other way around. It's now g prime or g of f of x. So once again, in order to find the derivative of h is going to require use of the chain rule. So starting on the outside, we'll start with g prime and we're going to keep that as f of x for now. Remember, it's one derivative at a time and now we need to multiply by the derivative of f of x, which simply is f prime of x. So remember we're trying to find h prime of two. So let's just fill in our twos for x's. So let's start on the inside right here. f of two and that is defined to be five. So it means on the next step, we're going to need g prime of five. f prime of two, that's e. So this part right there is e. Now going back to the table, g prime of five is seven. So we have seven times e as our answer. Now in this last one, we have the function h to be defined as f of f of x. So we start on the outside. So we have f prime of f of x, number one, derivative at a time. Now we multiply by the derivative of the inside function. So remembering that we want specifically to find h prime of two. So from the table and we'll start right in here at f of two. So that is equal to two. Oh that's five, I'm sorry. So we have f prime of five times, now f prime of two is e. So going back to the table once more, f prime of five now is pi. So we have pi times e as our answer for this.