d dx ( ^ - 1 1 + x^2 - 1 x ) is equal to - Vidyadip

MCQ

${d \over {dx}}\left( {{{\tan }^{ - 1}}{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$ is equal to

A
${1 \over {1 + {x^2}}}$
✓
${1 \over {2(1 + {x^2})}}$
C
${{{x^2}} \over {2\sqrt {1 + {x^2}} (\sqrt {1 + {x^2}} - 1)}}$
D
${2 \over {1 + {x^2}}}$

✓

Answer

Correct option: B.

${1 \over {2(1 + {x^2})}}$

b
(b) Let $y = {\tan ^{ - 1}}\frac{{\sqrt {1 + {x^2}} - 1}}{x}$

Put $x = \tan \theta ,$ then

$y = {\tan ^{ - 1}}\left( {\frac{{\sqrt {1 + {{\tan }^2}\theta } - 1}}{{\tan \theta }}} \right)$

$y = {\tan ^{ - 1}}\left( {\frac{{\sec \theta - 1}}{{\tan \theta }}} \right) = {\tan ^{ - 1}}\left( {\frac{{1 - \cos \theta }}{{\sin \theta }}} \right)$

$y = {\tan ^{ - 1}}\left( {\frac{{2{{\sin }^2}\frac{\theta }{2}}}{{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}}} \right) = {\tan ^{ - 1}}\tan \frac{\theta }{2}$

$y = \frac{\theta }{2} = \frac{1}{2}{\tan ^{ - 1}}x$, $(\because \theta ={{\tan }^{-1}}x)$

Hence $\frac{{dy}}{{dx}} = \frac{1}{{2(1 + {x^2})}}$.

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

$\mathop {\lim }\limits_{x \to 0} \,\left( {\frac{{x - \sin \,x}}{x}} \right)\,\sin \,\left( {\frac{1}{x}} \right)$

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{6} x+3=0$ such that $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Let $a, b$ be integers not divisible by $3$ and $n$ be a natural number such that $\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+i b), i=\sqrt{-1}$. Then $\mathrm{n}+\mathrm{a}+\mathrm{b}$ is equal to_.....................

If ${\sum\limits_{i = 1}^{20} {\left( {\frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3}\, = \frac{k}{{21}}$, then $k$ equals

Let $A$ and $B$ be subsets of a set $X$. Then

If the line $\frac{x}{a} + \frac{y}{b} = 1$ passes through the points $(2, -3)$ and $(4, -5)$, then $(a,\;b)$=

Find the values of $\sin \left(\frac{\pi}{3}-\sin ^{-1}\left(-\frac{1}{2}\right)\right)$ is equal to

Let $| X |$ denote the number of elements in set $X$. Let $S =\{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are indepenent events associated with $S$, then the number of ordered pairs $(A, B)$ such that $1 \leq|B|<|A|$, equals

${C_1} + 2{C_2} + 3{C_3} + 4{C_4} + .... + n{C_n} = $

Let $a, b, c, d$ be real numbers such that $\sum \limits_{k=1}^n\left(a k^3+b k^2+c k+d\right)=n^4$, for every natural number $n$. Then, $|a|+|b|+|c|+|d|$ is equal to

Let $f :\left[\frac{1}{2}, 1\right] \rightarrow R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{\prime}(x)<2 f(x)$ and $f\left(\frac{1}{2}\right)=1$. Then the value of $\int_{1 / 2}^1 f(x) d x$ lies in the interval