A) \[\cos \text{ec}\,x\]
B) \[-\cos \text{ec}\,x\]
C) \[\sec x\]
D) \[-\sec x\]
Correct Answer: C
Solution :
\[\frac{d}{dx}\log \tan \left( \frac{\pi }{4}+\frac{x}{2} \right)=\frac{1}{\tan \left( \frac{\pi }{4}+\frac{x}{2} \right)}{{\sec }^{2}}\left( \frac{\pi }{4}+\frac{x}{2} \right).\frac{1}{2}\] \[=\frac{1}{2}.\frac{1}{\sin \left( \frac{\pi }{4}+\frac{x}{2} \right)\cos \left( \frac{\pi }{4}+\frac{x}{2} \right)}=\frac{1}{\sin \left( \frac{\pi }{2}+x \right)}=\frac{1}{\cos x}=\sec x\].
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