Mark M. answered • 11/28/23
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We are given that h(x) = √5+4f(x)
Therefore the derivative of h is
h'(x) = 4f'(x)
Let's plug x=5 into the above.
h'(5) = 4f'(5) = 4 · 3 = 12
If I had a way of making a diagram here, I could show you why the derivative of h is 4 times the derivative of f.
In other words, h'(x) = 4f'(x)
Rael M. C.
tutor
did you mean the entire expression was inside of the square root?
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11/28/23
Mark M.
Rael M. C is right: if the entire expression 5 + 4f(x) is inside the square root sign, then h'(5) = 6/5. Here's why: h(x) = √5+4f(x) Let's write parentheses to show that the entire 5+4f(x) is under the square root: h(x) = √(5+4f(x)) Even better, let'ss write the square root as the exponent (superscript) .5: h(x) = (5 + 4f(x))^.5 Now we can use the Chain Rule: h'(x) = (.5)(5 + 4f(x))^-.5 times 4f'(x) (because .5 minus 1 is -.5) Plugging in x = 5, h'(5) = (.5)(5 + 4f(5))^-.5 times 4'(5) Doing the arithmetic, h'(5) = (.5)(5 + 4 times 5)^-.5 times 4 times 3 Simplifying, h'(5) = (.5)(25)^-.5 times 12 Simplifyig some more, h'(5) = (.5)(1/5) times 12 (note that 25 to the power negative half is 1/5) Simplifying some more, h'(5) = (1/10) times 12 Finally, h'(5) = 5/6
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11/28/23
Michael C.
It says it's incorrect and that it is 6/5 because of the chain rule but thank you.11/28/23