Question

$\frac{d}{dx}\left(x^3{\tan }^2\left(\frac{x}{2}\right)\right)=$

• A. $x^3\tan \left(\frac{x}{2}\right){\sec }^2\left(\frac{x}{2}\right)+3x{\tan }^2\left(\frac{x}{2}\right)$
• B. $x^3{\tan }^2\left(\frac{x}{2}\right){\sec }^2\left(\frac{x}{2}\right)+3x^2{\tan }^2\left(\frac{x}{2}\right)$
• C. $x^3\tan \left(\frac{x}{2}\right){\sec }^2\left(\frac{x}{2}\right)+3x^2{\tan }^2\left(\frac{x}{2}\right)$
• D. none of these

Answer 1

The correct option is C

$x^3\tan \left(\frac{x}{2}\right){\sec }^2\left(\frac{x}{2}\right)+3x^2{\tan }^2\left(\frac{x}{2}\right)$

Explanation for correct option:

Given, $\frac{d}{dx}\left(x^3{\tan }^2\left(\frac{x}{2}\right)\right)$

Apply theorem for differentiation of product of two or more function. $\frac{d}{dx}\left(u·v\right)=v\frac{du}{dx}+u\frac{dv}{dx}$

$$\begin{gathered} =3x^2{\tan }^2\left(\frac{x}{2}\right)+x^{3}2\tan \left(\frac{x}{2}\right)\times {\sec }^2\left(\frac{x}{2}\right)\times \frac{1}{2}\times 1 \\ =3x^2{\tan }^2\left(\frac{x}{2}\right)+x^3\tan \left(\frac{x}{2}\right){\sec }^2\left(\frac{x}{2}\right) \\ =x^3\tan \left(\frac{x}{2}\right){\sec }^2\left(\frac{x}{2}\right)+3x^2{\tan }^2\left(\frac{x}{2}\right) \end{gathered}$$

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