Question

The differential coefficient of sec $\left({\tan }^{-1}x\right)$

• A. $\frac{x}{(1+x^2)}$
• B. $x\sqrt{1+x^2}$
• C. $\frac{1}{\sqrt{1+x^2}}$
• D. $\frac{x}{\sqrt{1+x^2}}$

Answer 1

Answer: (A)

$\frac{x}{(1+x^2)}$

$\frac{d}{dx}\left(\sec \left({\tan }^{-1}x\right)\right)=\frac{d}{dx}\left(\sec \left({\sec }^{-1}\left(\sqrt{{1+x}^2}\right)\right)\right)$

$=\frac{d}{dx}\left(\sqrt{{1+x}^2}\right)=\frac{\frac{d}{dx}\left(\sqrt{1+x^2}\right)}{2\sqrt{{1+x}^2}}$
$=\frac{2x}{2\sqrt{{1+x}^2}}=\frac{x}{\sqrt{{1+x}^2}}$