Question

If $y =\log_{e}x \left(x-2\right)^{2}$ for $x \ne0 , 2$ then $y' \left(3\right) =$

• A. 44564
• B. 44595
• C. 44654
• D. None of these.

Answer 1

Answer: (B)

44595

Answer 2

$y =\log_{e}x \left(x-2\right)^{2} = \frac{\log\left(x-2\right)^{2}}{\log e^{x}}$ $= \frac{2\log\left(x-2\right)}{x \log e} = \frac{2\log\left(x-2\right)}{x}$ $\therefore \frac{dy}{dx} = 2 . \frac{x. \frac{1}{x-2} - \log\left(x-2\right)1}{x^{2}}$ $\therefore y'\left(3\right) = \frac{2}{9} \left(\frac{3}{1} - \log1\right) = \frac{2}{4} \left(3\right) = \frac{2}{3}$