$\begin{aligned} & y=\log \sqrt{\frac{1+\sin x}{1-\sin x}}=\frac{1}{2}\{\log (1+\sin x)-\log (1-\sin x)\} \\ & \left\{\frac{1}{1+\sin x} \cdot \cos x-\frac{1}{1-\sin x} \cdot(-\cos x)\right\} \\ & \Rightarrow \frac{d y}{d x}_{\text {at } x=\frac{\pi}{3}}=\frac{1}{2}\left\{\frac{1}{1+\sin \frac{\pi}{3}} \cos \frac{\pi}{3}-\frac{1}{1-\sin \frac{\pi}{3}} \cdot\left(-\cos \frac{\pi}{3}\right)\right\} \\ & =\frac{1}{2} \frac{1}{2}\left\{\frac{1}{1+\frac{\sqrt{3}}{2}} \cdot \frac{1}{2}+\frac{1}{1-\frac{\sqrt{3}}{2}} \cdot \frac{1}{2}\right\} \\ & =\frac{1}{2}\left\{\frac{1}{2+\sqrt{3}}+\frac{1}{2-\sqrt{3}}\right\} \\ & =\frac{1}{2} \times \frac{4}{4-3}=2 \\ & \end{aligned}$