Download App

5. continuity and differentiation — ગણિત ધોરણ 12 સાયન્સ — Question

Gujarat Boardગુજરાતી માધ્યમધોરણ 12 સાયન્સગણિત5. continuity and differentiation1 Mark
MCQ
${d \over {dx}}\left\{ {{{\cos }^{ - 1}}\left( {{{1 - {x^2}} \over {1 + {x^2}}}} \right)} \right\} = $
  • A
    ${1 \over {1 + {x^2}}}$
  • B
    $ - {1 \over {1 + {x^2}}}$
  • C
    $ - {2 \over {1 + {x^2}}}$
  • ✓
    ${2 \over {1 + {x^2}}}$
✓

Answer

Correct option: D.
${2 \over {1 + {x^2}}}$
d
(d) $\frac{d}{{dx}}\left\{ {{{\cos }^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right)} \right\}$

Let $\frac{{1 - {x^2}}}{{1 + {x^2}}} = \cos \theta $

==> $1 - {x^2} = (1 + {x^2})\cos \theta $

==> $ - {x^2}(1 + \cos \theta ) = \cos \theta - 1$

==> ${x^2} = \frac{{1 - \cos \theta }}{{1 + \cos \theta }} = \frac{{2{{\sin }^2}\frac{\theta }{2}}}{{2{{\cos }^2}\frac{\theta }{2}}} = {\tan ^2}\frac{\theta }{2}$

or $x = \tan \frac{\theta }{2}$ or $\theta = 2{\tan ^{ - 1}}x$

So, $\frac{d}{{dx}}[\theta ] = \frac{d}{{dx}}[2{\tan ^{ - 1}}x] = \frac{2}{{1 + {x^2}}}$.

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

More "MCQ" from 5. continuity and differentiation→5. continuity and differentiation — all question types→ગણિત ધોરણ 12 સાયન્સ question papers→Gujarat Board ધોરણ 12 સાયન્સ question papers→Gujarat Board question paper generator→

Similar questions

મુખ્ય કિંમત શોધો : $\cot ^{-1}(\sqrt{3})$
→
$\cos ({\tan ^{ - 1}}(\tan 2)) =\ .... . . .$
→
$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x^3+x \cos x+\tan ^5 x+1\right) d x$ નું મૂલ્ય
→
જો $a,b,c$ ધન અને અસમાન હોય , તો નિશ્રાયક $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right|$ ની કિમત $ . ....... .$
→
One ticket is selected at random from $50$ tickets numbered $00,01,02, ... ,49.$ Then the probability that the sum ofthe digits on the selected ticket is $8$, given that the product of these digits is zero, equals:
→
જો $A = \left[ {\begin{array}{*{20}{c}}
{ - 4}&{ - 1}\\
3&1
\end{array}} \right]$ , તો શ્રેણિક $\left( {{A^{2016}} - 2{A^{2015}} - {A^{2014}}} \right)$ ના નિશ્રાયકની કિમંત મેળવો.
→
$\mathop {\lim }\limits_{n \to \infty } \frac{{1 + {2^4} + {3^4} + .... + {n^4}}}{{{n^5}}}$$ - \mathop {\lim }\limits_{n \to \infty } \frac{{1 + {2^3} + {3^3} + .... + {n^3}}}{{{n^5}}} = $
→
$\sec ^2\left(\tan ^{-1} 2\right)+\operatorname{cosec}^2\left(\cot ^{-1} 2\right)=\ .......... $
→
$F(x)=\frac{x^3}{3}+\frac{3x^3}{2}+ax+b.$ જો $F(-2)=0$ હોય તો એવી કેટલી ક્રમયુક્ત જોડ $(a,b)$ શક્ય બને જયા વિધેય $F$ એ $P(-2,0)$ આગળ ન્યુનતમ હોય.
→
અહી $f : R \rightarrow R$ એ વિકલનિય વિધેય છે કે જેથી $f^{\prime}(x)+f(x)=\int \limits_0^2 f(t) d t$ થાય જો  $f(0)=e^{-2}$ હોય તો  $2 f (0)- f (2)$ ની કિમંત  $.........$ મેળવો.
→