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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 is equal to lowercase G of f of X 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 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 side 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 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 store a 2.5 in there so let's try to do that so G prime 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 lowercase G prime of F of F of 2.5 F of 2.5 x times F prime of 2.5 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 I'm just in 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 write it this way F prime 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 way it's actually a line and the slope is very 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 we see that for every for every 3 that we run we go up to or that we rise to for every 3 that we run or that our change in Y over change in X is 2 over 3 so the slope 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 over 1 if we go one in the horizontal direction we go up 2 in the vertical direction change in Y over change in X is 2 over 1 so G Prime let me write this down G prime of 1 G prime of 1 is equal to 2 is equal to 2 so this whole thing evaluates to 2 and so this simplifies we scratch that out this simplifies to 2 times 2/3 which is equal to 4/3 so we could write G prime of 2.5 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