求y = ln tan (x)/(2)的微分.

设 $ y = \ln\left(\tan\left(\frac{x}{2}\right)\right) $，对 $ x $ 求导得 \[ \frac{dy}{dx} = \frac{1}{\tan\left(\frac{x}{2}\right)} \cdot \frac{d}{dx}\left(\tan\left(\frac{x}{2}\right)\right) = \frac{1}{\tan\left(\frac{x}{2}\right)} \cdot \frac{1}{2}\sec^2\left(\frac{x}{2}\right) \] 利用恒等式 $ \sec^2\left(\frac{x}{2}\right) = \frac{1}{\cos^2\left(\frac{x}{2}\right)} $ 和 $ \tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)} $，化简得 \[ \frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)} = \frac{1}{\sin(x)} = \csc(x) \] 因此，微分 $ dy $ 为 \[ \boxed{\csc(x) \, dx} \] （或等价表示：$\boxed{\frac{dx}{\sin(x)}}$）