 Let's take a look at how we can use the chain rule to find requested derivatives when information is given to us in the form of a graph. So here we have two graphs of functions. The red graph is function f, the blue graph is function g, and you'll notice in the instructions there we have three other functions defined as compositions of those two functions f and g. We have a u-function, a v-function, and a w-function, and in the next three problems, we're going to be asked to find those derivatives evaluated, I believe, at the x-value of 1. So the first one we are asked to find is the derivative of u at 1, and I have the graph there replicated for you. Remember that it was given to us earlier that the u-function was the composite function f-composed g, f of g of x. So if we want to find u prime of 1, that's going to require use of the chain rule. So let's just set it up as to what the derivative would look like. Remember we start on the outside and work our way in, then we have to go inside and multiply by the derivative of g. So that's just the statement of the chain rule. Now we specifically want to find the derivative at 1. So I'm just going to fill in 1 here for our x's, and then the rest of the information we should be able to get from the graph. Now starting in the square brackets, g of 1, that is just the function value of g when x equals 1, which looks like it's 3 in this case if you take a look at the graph. So that means we have so far f prime of 3. Now the next part we need right here is g prime of 1. So since we're given a graph, really what that's asking you is at the point where x equals 1 on the graph, what is the slope of that g function? So if you take a look at it, you'll find that it's equal to negative 3. You can give down 3 over 1, down 3 over 1, so that's negative 3. So going back to our graph, we now need f prime of 3. So that means at the x value of 3, we need to know the slope of that f function. So probably the easiest thing to do is maybe count down over. It looks like it's going to be 1 fourth, a negative 1 fourth. So the slope at 3 on the graph, of course 3 is right here. No, that's not 3, that's 4. There's 3 right there. You can figure out the slope easily of that part of the f graph and it's negative 1 fourth. So we have that times a negative 3, so our answer in the end is 3 fourths. So let's take a look at another one, this time involving the v function. So we still want to find v prime of 1. The v function was given to be the composite function g of f of x. So if we just go ahead and find our derivative by using the chain rule starting on the outside, taking one derivative at a time. So this then would simply be the statement of the chain rule itself. Now we want the derivative at 1. So let's start in square brackets. We need the function value of f at 1, which looks to be 2. So so far we're looking at g prime of 2. Now we need f prime of 1. So that means right here where 1 is. On the f graph we need the slope there, so you could count up and over. It ends up being up to over 1, up to over 1, so the derivative there is 2. Now we need to do something similar at 2 on the g graph. Now notice what's happening on the g graph where x equals 2. We have a cusp. Remember the derivative does not exist there. So that sort of kills the whole problem because of that cusp point at x equals 2 on the g graph. We are not able to find the derivative there. So let's look at one more involving this time, the w function. Now the w function was given to be the composite function g of x. So when we go to find the derivative using the chain rule, starting on the outside, inside one derivative at a time, remember? So we want specifically the derivative at 1. So let's start in the square brackets at g of 1. So that value looks to be 3 from what I can tell from the graph. Right, because here of course is 1. And you're looking at the blue graph, so the function value there looks to be 3. Now for g prime of 1 what we need is the slope there. We talked about this earlier. So that's going to be negative 3. If you think down 3 over 1, down 3 over 1. Now we need g prime of 3. So here's 3. We need the slope of that segment of the g function where x equals 3. So again maybe think up 2 in this case over 3. So the slope is 2 thirds. So we have 2 thirds times the negative 3. So our answer in the end is negative 2. So hopefully this serves as a good example for you of how you can again really have to understand how the chain rule works and then how to read a graph and interpret that and piece the two together in order to find derivatives that are requested of you.