What is d/dx(logtanx/2) for 0 < x < π?

Question 79

HardOriginal question

What is $\frac{d}{dx}\left(\log\tan\frac{x}{2}\right)$ for $0 < x < \pi$ ?

2\operatorname{cosec} x

Correct answerYour answer

\frac{1}{2}\sec^{2}\frac{x}{2}

Correct answerYour answer

\operatorname{cosec} x

Correct answerYour answer

\cot\frac{x}{2}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$\frac{u'}{u}$ with $u = \tan\frac{x}{2}$ , $u' = \frac{1}{2}\sec^{2}\frac{x}{2}$ .
$\frac{\sec^{2}(x/2)}{2\tan(x/2)} = \frac{1}{2\sin\frac{x}{2}\cos\frac{x}{2}} = \frac{1}{\sin x} = \operatorname{cosec} x$ .

Worth noting

$2\operatorname{cosec} x$ forgets the factor $\frac{1}{2}$ from the inner $\frac{x}{2}$ .

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