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Problem 61 Medium Difficulty

If $ F(x) = f(g(x)), $ where $ f(-2) = 8, f'(-2) =4, f'(5) = 3, g(5) = -2, $ and $ g'(5) = 6, $ find $ F'(5). $

Answer

$F^{\prime}(5)=24$

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Video Transcript

Capital F of X is a composite function and we want capital f prime of five. So let's start by finding capital f prime of X, we're going to need to use the chain rule. So first we have the derivative of the outside function little f prime of G of X and then that's multiplied by the derivative of the inside function G prime of X. Now we can substitute five in for X and we have capital F Prime of five is equal to F prime of G 05 times g prime of five. Now we can substitute in the values we were given. We need to know G of five and g 05 is negative two so we can substitute that in. And now we have f prime of negative too. And we also need to know g prime of five and that is six. So we can substitute that in time. Six. Now we need to know f prime of negative too. And we do we know that that is four, so we can substitute that in. And so we have four times six. So the answer is 24