d dx ( x)^x = - Vidyadip

MCQ

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

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

✓

Answer

Correct option: B.

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

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

$ \Rightarrow \frac{{dy}}{{dx}} = {(\sin x)^x}[x\cot x + {\log _e}\sin x]$

$ = {(\sin x)^x}\left[ {\frac{{x\cos x + \sin x\log \sin x}}{{\sin x}}} \right]$.

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