Worked example: Chain rule with table

the following table lists the values of functions F and G and of their derivatives F Prime and G prime for the x-values negative 2 & 4 and so you can see 4x equals negative 2x equals 4 they give us the values of F G F Prime and G Prime let function capital F be defined as the composition of F and G it's lowercase F of G of X and they want us to evaluate F prime of 4 so you might immediately recognize that if I have a function that can review it as a composition of other functions that the chain rule will apply here and so and I'm just going to restate the chain rule the derivative of capital F is going to be the derivative of lowercase F the outside function with respect to the inside function so lowercase F prime of G of x times the derivative of the inside function with respect to X times G prime of X and if we're looking for F prime of 4 F prime of 4 well everywhere we see an X we replace it with a 4 that's going to be lowercase F prime of G of 4 times G prime of 4 now how do we figure this out they haven't given us explicitly the values of the functions on for all X's but they've given it to us at some interesting points so the first thing you might want to figure out is well what is G of 4 going to be well they tell us when X is equal to 4 G of 4 is negative 2 this tells us that the value that G of X takes on when X is equal to 4 is negative 2 so this right over here is negative 2 and so this first part is f prime of negative 2 so what is f prime what is f prime of negative 2 well when X is equal to negative 2 F prime is equal to 1 so this right over here is f prime of negative 2 that is equal to 1 and now we just have to figure out what G prime of 4 is well when let me circle this G prime of 4 when X is equal to 4 and I'll scroll down a little bit when X is equal to 4 prime takes on the value eight so there you have it F prime of 4 is equal to 1 times 8 which is equal to 8 and we're done