Answer: \(\dfrac{dy}{dx}=\dfrac23\) when \(x=\dfrac{\pi}{3}\).

We start with the main method. Use the product, chain and standard trigonometric derivative rules carefully.

Given

\(y=\tan\frac{x}{2}.\)

Differentiate using the chain rule:

\(\frac{dy}{dx}=\frac12\sec^2\frac{x}{2}.\)

When \(x=\frac{\pi}{3}\),

\(\frac{x}{2}=\frac{\pi}{6}.\)

So

\(\frac{dy}{dx}=\frac12\sec^2\frac{\pi}{6}.\)

Since \(\cos\frac{\pi}{6}=\frac{\sqrt3}{2}\),

\(\sec^2\frac{\pi}{6}=\frac{1}{\cos^2(\pi/6)}=\frac{1}{3/4}=\frac43.\)

Therefore

\(\frac{dy}{dx}=\frac12\cdot\frac43=\frac23.\)

This completes the solution and gives the required result.