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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Differential coefficient of  ${\sin ^{ - 1}}x$ $w.r.t$ ${\cos ^{ - 1}}\sqrt {1 - {x^2}} $ is
  • ✓
    $1$
  • B
    ${1 \over {1 + {x^2}}}$
  • C
    $2$
  • D
    None of these
✓

Answer

Correct option: A.
$1$
a
(a) Let ${y_1} = {\sin ^{ - 1}}x$ and ${y_2} = {\cos ^{ - 1}}\sqrt {1 - {x^2}} $

Differentiating w.r.t. $x$ of ${y_1}$ and ${y_2}$, we get

$\frac{{d{y_1}}}{{dx}} = \frac{1}{{\sqrt {1 - {x^2}} }}$

$\frac{{d{y_2}}}{{dx}} = - \frac{1}{{\sqrt {1 - (1 - {x^2})} }}\frac{{1( - 2x)}}{{2\sqrt {1 - x} }} $

$= \frac{1}{{\sqrt {1 - {x^2}} }} \Rightarrow \frac{{d{y_2}}}{{d{y_1}}} = 1.$

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