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Numerade Educator

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Problem 62 Hard Difficulty

If $ h(x) = \sqrt {4 + 3f(x)}, $ where $ f(1) = 7 $ and $ f'(1) = 4, $ find $ h'(1). $

Answer

$\frac{6}{5}$

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Video Transcript

all right. H of X is a composite function, and we want to find aged prime of one. So let's start by finding h prime of X. And to do that, we need to use the chain rule, and I'm actually going to start by rewriting h of x as four plus three f of X to the 1/2 power. So now, using the chain rule, the derivative is 1/2 times four plus three f of X to the negative 1/2 power times the derivative of the inside and the derivative of 40 and the derivative of three F of X would be three f prime of X. Now let's evaluate that at X equals one. So age prime of one is going to be 1/2 times four plus three times f of one to the negative 1/2 power times three times f prime of one. So we need to substitute f of one, which is seven, and we need to substitute f prime of one, which is four. So we have aged prime of one equals 1/2 times four plus three times seven to the negative, 1/2 power times three times For now, it's simplify that four plus three times seven that would be 25 and three times for that would be 12 25 to the negative 1/2 power is one over the square root of 25 so that would be 1/5. So we have 1/2 times 1/5 times 12 and that gives us 6/5.