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Problem 44 Hard Difficulty

Suppose that $ f(4) = 2, g(4) = 5, f'(4) = 6. $ and $ g'(4) = -3. $ Find $ h'(4). $

(a) $ h(x) = 3f(x) + 8g(x) $

(b) $ h(x) = f(x) g(x) $

(c) $ h(x) = \frac {f(x)}{g(x)} $

(d) $ h(x) = \frac {g(x)}{f(x) + g(x)} $

Answer

(a) -6
(b) 24
(c) $\frac{36}{25}$
(d) $-\frac{36}{49}$

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Video Transcript

it's close. So when you read here So for part A, we have each the derivative. It was equal to three times the derivative of plus eight times the derivative of G. This becomes equal to you. Plug in four for X and you get three times six plus eight times negative three, which becomes equal to negative six. Report be Each is a product of F N G. So we get the derivative of H. It is equal to X times the derivative of G less g of X runs the derivative of we're gonna plug in four for X when we get equal to two times negative three list by I'm six is equal to 24 For part C, we have a TSH is a quotient of f N g. So the derivative is equal to G times the derivative minus half times the derivative of G all over G square. This becomes equal to the derivative of each of four is equal to G of four terms, the derivative of F A four minus up before times the derivative of G of four, all over G of four square, which is equal to 36 over 25 for party we have you of X. We're gonna make that equal F X plus g of vets. So you Is this some of f n g? Thank you. The derivative of you is equal to when you add the derivative of us I n g each is a quotient of g of X and you have X So we get each of X g over you. The derivative is equal to you. How's the derivative of G minus G tons The derivative of you all over. You swear I'll continue here we get a tch, the derivative of X, and this is equal to f of X plus g of nuts. Times the derivative of G minus G of X times the derivative of X plus, the derivative of G of X, all over ever backs less g of x square. When we put this in, we got two plus four two plus five excusing arms negative three minus by I'm six minus three because you're adding negative three and he divide this over to plus five square to get 36 over 49 negative