A) \[{{x}^{3}}\tan \frac{x}{2}.{{\sec }^{2}}\frac{x}{2}+3x{{\tan }^{2}}\frac{x}{2}\]
B) \[{{x}^{3}}\tan \frac{x}{2}.{{\sec }^{2}}\frac{x}{2}+3{{x}^{2}}{{\tan }^{2}}\frac{x}{2}\]
C) \[{{x}^{3}}{{\tan }^{2}}\frac{x}{2}.{{\sec }^{2}}\frac{x}{2}+3{{x}^{2}}{{\tan }^{2}}\frac{x}{2}\]
D) None of these
Correct Answer: B
Solution :
\[\frac{d}{dx}\left( {{x}^{3}}{{\tan }^{2}}\frac{x}{2} \right)=3{{x}^{2}}{{\tan }^{2}}\frac{x}{2}+2{{x}^{3}}\tan \frac{x}{2}{{\sec }^{2}}\frac{x}{2}.\frac{1}{2}\] \[={{x}^{3}}\tan \frac{x}{2}{{\sec }^{2}}\frac{x}{2}+3{{x}^{2}}{{\tan }^{2}}\frac{x}{2}\].
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