d dx [ e^x ]= - Vidyadip

MCQ

${d \over {dx}}\left[ {\log \sqrt {\sin \sqrt {{e^x}} } } \right]=$

✓
${1 \over 4}{e^{x/2}}\cot ({e^{x/2}})$
B
${e^{x/2}}\cot ({e^{x/2}})$
C
${1 \over 4}{e^x}\cot \,({e^x})$
D
${1 \over 2}{e^{x/2}}\cot \,({e^{x/2}})$

✓

Answer

Correct option: A.

${1 \over 4}{e^{x/2}}\cot ({e^{x/2}})$

a
(a) $\frac{d}{{dx}}[\log \sqrt {\sin \sqrt {{e^x}} } ] = \frac{d}{{dx}}\left[ {\frac{1}{2}\log (\sin \sqrt {{e^x}} )} \right]$

$ = \frac{1}{2}\cot \sqrt {{e^x}} \frac{1}{{2\sqrt {{e^x}} }}{e^x} = \frac{1}{4}{e^{x/2}}\cot ({e^{x/2}})$

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

The mean and standard deviation of $40$ observations are $30$ and $5$ respectively. It was noticed that two of these observations $12$ and $10$ were wrongly recorded. If $\sigma$ is the standard deviation of the data after omitting the two wrong observations from the data, then $38 \sigma^{2}$ is equal to$.........$

The equation of perpendicular bisectors of the sides $AB$ and $AC$ of a triangle $ABC$ are $x - y + 5 = 0$ and $x + 2y = 0$ respectively. If the point $A$ is $(1,\; - \;2)$, then the equation of line $BC$ is

Consider a function $f : N \rightarrow R$, satisfying $f(1)+2 f(2)+3 f(3)+\ldots+x f(x)=x(x+1) f(x) ; x \geq 2$ with $f(1)=1$. Then $\frac{1}{f(2022)}+\frac{1}{f(2028)}$ is equal to

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ from any of its foci?

The number of real values of the parameter $k$ for which ${({\log _{16}}x)^2} - {\log _{16}}x + {\log _{16}}k = 0$ with real coefficients will have exactly one solution is

The value of $\mathop {\lim }\limits_{x \to - 1} \frac{{{x^2} + 3x + 2}}{{{x^2} + 4x + 3}}$ is equal to

The following system of equations $3x - 7y + 5z = 3; 3x + y + 5z = 7$ and $2x + 3y + 5z = 5$ are

$\int_{}^{} {\left[ {\log (\log x) + \frac{1}{{{{(\log x)}^2}}}} \right]} \;dx = $

If $\tan (x + y) = 33$ and $x = {\tan ^{ - 1}}3,$ then $y $ will be

The ellipse $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes.

Another ellipse $E _2$ passing through the point $(0,4)$ circumscribes the rectangle $R$.. The eccentricity of the ellipse $E _2$ is