Question

If $y={\sin }^{-1}\{\sqrt{x-ax}-\sqrt{a-ax}\}$, then $\frac{dy}{dx}$ is equal to

• A. $\frac{1}{\sin \sqrt{a-ax}}$
• B. $\sin \sqrt{x}\sin \sqrt{a}$
• C. $\frac{1}{2\sqrt{x(1-x)}}$
• D. $0$

Answer 1

The correct option is C $\frac{1}{2\sqrt{x(1-x)}}$

$y={\sin }^{-1}(\sqrt{a-ax}-\sqrt{a-ax})$
$\Rightarrow \frac{dy}{dx}=\frac{\frac{(1-a)}{2\sqrt{x-ax}}-\frac{(0-a)}{2\sqrt{a-ax}}}{\sqrt{1-{(\sqrt{x-ax}-\sqrt{a-ax})}^2}}$
$=\frac{2(1-a)\sqrt{a-ax}+2a\sqrt{x-ax}}{\sqrt{1-{\left(\sqrt{x-ax}\right)}^2-{\left(\sqrt{a-ax}\right)}^2+2\sqrt{a-ax}\sqrt{x-ax}}}$
$\times \frac{1}{\sqrt{(x-ax)(a-ax)}}=\frac{1}{2\sqrt{x(1-x)}}$