Question

Differential coefficient of $\sqrt{\sec \left(\sqrt{x}\right)}$ is

• A. $\left(\frac{1}{4\sqrt{x}}\right)\left(\sec \left(\sqrt{x}\right)\right)\left(\sin \left(\sqrt{x}\right)\right)$
• B. $\left(\frac{1}{4\sqrt{x}}\right){\left(\sec \left(\sqrt{x}\right)\right)}^{\frac{3}{2}}\left(\sin \left(\sqrt{x}\right)\right)$
• C. $\left(\frac{1}{2}\right)\sqrt{x}\left(\sec \left(\sqrt{x}\right)\right)\sin \left(\sqrt{x}\right)$
• D. $\left(\frac{1}{2}\right)\sqrt{x}{\left(\sec \left(\sqrt{x}\right)\right)}^{\frac{3}{2}}\sin \left(\sqrt{x}\right)$

Answer 1

The correct option is B

$\left(\frac{1}{4\sqrt{x}}\right){\left(\sec \left(\sqrt{x}\right)\right)}^{\frac{3}{2}}\left(\sin \left(\sqrt{x}\right)\right)$

Explanation for the correct option:

Let, the given, $y=\sqrt{\sec \left(\sqrt{x}\right)}$.

Differentiate with respect to $x$.

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{1}{2\sqrt{\sec \left(\sqrt{x}\right)}}\times \sec \left(\sqrt{x}\right)\tan \left(\sqrt{x}\right)\times \frac{1}{2\sqrt{x}} \\ \Rightarrow \frac{dy}{dx}=\frac{1}{4\sqrt{x}}\times {\sec }^{\frac{1}{2}}\left(\sqrt{x}\right)\times \frac{\sin \left(\sqrt{x}\right)}{\cos \left(\sqrt{x}\right)} \\ \Rightarrow \frac{dy}{dx}=\frac{1}{4\sqrt{x}}\times {\sec }^{\frac{3}{2}}\left(\sqrt{x}\right)\times \sin \left(\sqrt{x}\right) \end{gathered}$$

Hence, correct answer is option $\left(B\right)$.