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Let $ r(x) = f (g(h(x))), $ where $ h(1) = 2, g(2) = 3, h'(1) = 4, g'(2) = 5, $ and $ f'(3) = 6. $ Find $ r'(1). $

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Baylor University

University of Michigan - Ann Arbor

University of Nottingham

our goal is to find our prime of one. We have R of X. It's a composite function. So we're going to use the chain rule and let's start by finding our prime of X. So we start with the derivative of the outside function F with F prime of G of h of X. Then we moved to the middle function G, and we have its derivative g prime of h of X. Then we moved to the inside function H. And it's derivative is age prime of X. Now we want to substitute one in here so our prime of one would be f prime of G of age of one times g, prime of age of one times h, prime of one. Now we can start substituting some of the numbers in here. H of one is to we get that again and then we get h prime of one is for So when we substitute those in, we have f prime of G of to times G prime of to times four. Now we need G of to G of two is three and when you g prime of two g prime of two is five So now we have f prime of three times. Five times four. Okay, One more step to go. We need F prime of three f prime of three of six. So now we have six times, five times four and that's 1 20

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