d dx ( x)^ x = - Vidyadip

MCQ

${d \over {dx}}\{ {(\sin x)^{\log x}}\} = $

A
${(\sin x)^{\log x}}\left[ {{1 \over x}\log \sin x + \cot x} \right]$
✓
${(\sin x)^{\log x}}\left[ {{1 \over x}\log \sin x + \cot x\log x} \right]$
C
${(\sin x)^{\log x}}\left[ {{1 \over x}\log \sin x + \log x} \right]$
D
None of these

✓

Answer

Correct option: B.

${(\sin x)^{\log x}}\left[ {{1 \over x}\log \sin x + \cot x\log x} \right]$

b
(b) Let $y = {(\sin x)^{\log x}} $

$\Rightarrow {\log _e}y = {\log _e}x{\log _e}\sin x$

==> $\frac{{dy}}{{dx}} = {(\sin x)^{{{\log }_e}x}} = \left[ {\frac{1}{x}{{\log }_e}\sin x + \cot x{{\log }_e}x} \right]$.

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