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]$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If the shortest distance between the line $\vec{r}=(-\hat{i}+3 \hat{k})+\lambda(\hat{i}-a \hat{j})$ and $\vec{r}=(-\hat{j}+2 \hat{k})+\mu(\hat{i}-\hat{j}+\hat{k})$ is $\sqrt{\frac{2}{3}}$, then the integral value of $a$ is equal to

A natural number is selected at random from the set $\{1 \leq x \leq 100\}.$ The probatility that number satisfies the inequation $x^2 -13x \leq 30$ is :-

Which of the following is a true statement

For the differentiable function $f: R -\{0\} \rightarrow R$, let $3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10$, then $\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|$ is equal to

The value of $cot\, x + cot\, (60^o + x) + cot\, (120^o + x)$ is equal to :

The area of the region bounded by the lines $x=1, x=2$, and the curves $x\left(y-e^x\right)=\sin x$ and $2 x y=2 \sin x+x^3$ is

$\int_{}^{} {\frac{{\log x}}{{{{(1 + \log x)}^2}}}dx = } $

If the variance of the terms in an increasing $A.P.$, $b _{1}, b _{2}, b _{3}, \ldots b _{11}$ is $90,$ then the common difference of this $A.P.$ is

The length of the longest interval, in which the function $3\sin x - 4{\sin ^3}x $ is increasing, is

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{1 + x}&1\\1&1&{1 + y}\end{array}\,} \right| = $