Question

$\frac{d}{dx}\left(x\sqrt{a^{2}-x^{2}}+a^{2}\,sin^{-1}\left(\frac{x}{a}\right)\right)$ is equal to

• A. $\sqrt{a^{2}-x^{2}}$
• B. $2\sqrt{a^{2}-x^{2}}$
• C. $\frac{1}{\sqrt{a^{2}-x^{2}}}$
• D. none of these

Answer 1

Answer: (B)

$\frac{d}{dx}\left\{x\sqrt{a^{2}-x^{2}}+a^{2}\,sin^{-1}\left(\frac{x}{a}\right)\right\}$

$=\frac{x \times 1 \times \left(-2x\right)}{2\sqrt{a^{2}-x^{2}}}+\sqrt{a^{2}-x^{2}}+a^{2} \frac{1}{\sqrt{1-\frac{x^{2}}{a^{2}}}} \times\frac{1}{a}$
$=\frac{-x^{2}}{\sqrt{a^{2}-x^{2}}}+\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{a^{2}-x^{2}}$
$=\sqrt{a^{2}-x^{2}}+\frac{\left(a^{2}-x^{2}\right)}{\sqrt{a^{2}-x^{2}}}$
$=2\sqrt{a^{2}-x^{2}}$