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Numerade Educator

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Problem 64 Hard Difficulty

Let $ f $ and $ g $ be the function in Exercise 63.
(a) If $ F(x) = f(f(x)), $ find $ F'(2). $
(b) If $ G(x) = g(g(x)), $ find $ G'(3). $

Answer

a. 20
b. 63

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Video Transcript

in this problem. Capital F of X is f of f of X and we want to find capital f prime of to. So let's find capital f prime of X using the chain rule The derivative of the outside would be f prime of f of x times, the derivative of the inside F prime of X. Now let's evaluate that at two. So capital F prime of two would be f prime of f of to times f prime of to Now we use the table to find f of to f of two is one and f prime of to if prime of two is five and we substitute those numbers in So we have f prime of one times five. Now we go back to the table to find f prime of one of prime of oneness four Suite four times five. So the answer is 20. And for part B, we have capital G FX, which is G of G of X and we want to find capital G prime of three. So let's find capital G prime of X using the chain rule. The derivative of the outside would be g prime of G of X Times, the derivative of the inside G prime of X. Now let's evaluate that at three. So G prime of three would be g prime of G of three times g Prime of three. Now let's use the table to find G of three. G of three is too. And to find g prime of three g Prime of three is nine so we can substitute those values in and we get G prime of to times nine. Now we can look back at the table for G Prime of two g Prime of two is seven to have seven times nine, so the answer is 63.