For x2−4≠0, the value of ddxlog⁡exx−2x+23/4 at x=3 is

ddxlog⁡exx−2x+23/4=ddxx+34log⁡(x−2)−34log⁡(x+2)=1+34(x−2)−34(x+2)=1+34−320=20+15−320=3220=85 at x=3

For x2−4≠0, the value of ddxlog⁡exx−2x+23/4 at x=3 is

ddxlog⁡exx−2x+23/4=ddxx+34log⁡(x−2)−34log⁡(x+2)=1+34(x−2)−34(x+2)=1+34−320=20+15−320=3220=85 at x=3

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For $x^{2} - 4 \neq 0$ , the value of $\frac{d}{dx} [\log \{e^{x} {(\frac{x - 2}{x + 2})}^{3 / 4}\}] at x = 3 is$

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$8 e^{3} / 5$

8/5

answer is B.

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Detailed Solution

$\begin{array}{l} \frac{d}{dx} [\log \{e^{x} {(\frac{x - 2}{x + 2})}^{3 / 4}\}] = \frac{d}{dx} [x + \frac{3}{4} \log (x - 2) - \frac{3}{4} \log (x + 2)] = 1 + \frac{3}{4 (x - 2)} - \frac{3}{4 (x + 2)} \\ = 1 + \frac{3}{4} - \frac{3}{20} = \frac{20 + 15 - 3}{20} = \frac{32}{20} = \frac{8}{5} at x = 3 \end{array}$