Let $f(x)=(1+\sqrt{x})(2x^{2}-3)$. Then $f'(4)=$

Question

BlackTom AI · Accepted Answer

We start by restating the problem: given f(x) = (1 + √x)(2x^2 − 3), find f'(4).
First, identify the product structure f(x) = g(x)·h(x) with g(x) = 1 + √x and h(x) = 2x^2 − 3.
Compute derivatives: g'(x) = d/dx(1 + √x) = d/dx(√x) = 1/(2√x), and h'(x) = d/dx(2x^2 − 3) = 4x.
Apply the product rule: f'(x) = g'(x)·h(x) + g(x)·h'(x).
Now substitute x = 4:
- √4 = 2, so g(4) = 1 + 2 = 3.
- g'(4) = 1/(2·√4) = 1/(2·2) = 1/4.
- h(4) = 2·(4)^2 − 3 = 2·16 − 3 = 32 − 3 = 29.
- h'(4) = 4·4 = 16.
Compute f'(4) = (1/4)·29 + 3·16 = 29/4 + 48.
Convert 48 to quarters: 48 = 192/4, so f'(4) = (29 + 192)/4 = 221/4.
Thus, the derivative at x = 4 is 221/4.
In the context of the given multiple-choice answer, the value 221/4 corresponds to option labeled as b. 221/4. Note: the provided data shows the correct numerical result, but the list of answer choices is missing in the prompt, so the alignment to a specific letter choice relies on the stated correspondence in the answer field.
If you had to choose among typical options, any choice equaling 221/4 would be correct, and others would be incorrect due to either arithmetic errors or misapplication of the product rule.

SCNC1111_2B_2024 Quiz 5

Let \(f(x)=(1+\sqrt{x})(2x^{2}-3)\). Then \(f'(4)=\)

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