Differential coefficient of ^ - 1 x w.r.t ^ - 1 1 - x^2 is - Vidyadip

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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