Applying the chain rule graphically 1 (old)

Consider the functions f and g with the graphs shown below. If capital G of x is equal to lowercase g of lowercase f of x, what is the value of capital G prime of 2.5.? So G of x is a composition of g and f. So it's g of f of x, or lowercase g of f of x. And they don't graph capital G of x here. They just give us the graphs of lowercase g of x and lowercase f of x. This is the graph of lowercase f of x. This is a graph of lowercase g of x. So let's just try to think how we could evaluate this and then see if they've given us the right information here. So let me just rewrite a lot of what they've already told us. They've already told us that capital G of x is equal to lowercase g of f of x. So if we wanted to take the derivative of capital G of x-- and we do want to think about what the derivative of capital G of x is, because they want us to evaluate the derivative at x is equal to 2.5. So let's do that. Let's take the derivative of both sides of this. So if we take the derivative of the left hand side, we end up with G prime of x. And on the derivative on the right hand, since we have a composition here of two functions, we would apply the chain rule. So this is going to be the derivative of g with respect to f. So we could write that as g prime of f of x times the derivative of f with respect to x. So times f prime of x. So if we want to evaluate what G prime of 2.5 is, then every place we see an x here, we have to start with 2.5 in there. So let's try to do that. So G prime of-- and I'll do this in white so it sticks out-- G prime of 2.5 is going to be equal to lowercase g prime of f of 2.5 times f prime of 2.5. So let's think about what these would evaluate to. What is f of 2.5? Well, when x is equal to 2.5-- let me just get a color you can actually see. When x is equal to 2.5, our function here is equal to 1. So f of 2.5, so we know that f of 2.5 is equal to 1. Let me write that down, f of 2.5 is equal to 1. And we also need to figure out what f prime of 2.5 is. So f-- let me write it this way-- f prime of 2.5 is equal to. Now what is f prime of 2.5? That's just essentially the slope of the tangent line at the function when x is equal to 2.5. So it's really just the slope right over here. And at least right over at this part of the function, it's actually a line. And the slope is actually very easy to spot out. If we were to go from this point to this point here-- and I'm just picking those points because those are on kind of integer-valued coordinates-- we see that for every three that we run, we go up two, or that we rise two for every three that we run. Or that our change in y over change in x is 2/3. So the slope of the function right over there is 2/3. So this is equal to 2/3. And so we can substitute back in here, f of 2.5 is equal to 1. And this right over here is equal to 2/3. Now, we're not done yet. Now we have to evaluate, what is G prime of 1? So when x is equal to 1, this is the function g. We're not evaluating g of 1. We're evaluating g prime of 1. So what is the slope of the line here? Well, our change in y over change in x is 2/1. If we go one in the horizontal direction, we go up two in the vertical direction. Change in y over change in x is 2/1. So g prime-- let me write this down-- g prime of 1 is equal to 2. So this whole thing evaluates to 2. And so this simplifies-- let me scratch that out-- the simplifies to 2 times 2/3, which is equal to 4/3. So we could write G prime of 2.5 is equal to 4/3. And this is a pretty neat problem, because we didn't get to see the actual function definition from G of x. But just using the chain rule and the information they're giving us, we were able to figure out what the value of this derivative is when x is equal to 2.5.