(i) Let \(y = \ln(1 + \sin 2x)\). Using the chain rule, \(\frac{dy}{dx} = \frac{1}{1 + \sin 2x} \cdot \frac{d}{dx}(1 + \sin 2x)\).

The derivative of \(1 + \sin 2x\) is \(2 \cos 2x\). Therefore, \(\frac{dy}{dx} = \frac{2 \cos 2x}{1 + \sin 2x}\).

(ii) Let \(y = \frac{\tan x}{x}\). Using the quotient rule, \(\frac{dy}{dx} = \frac{x \cdot \sec^2 x - \tan x \cdot 1}{x^2}\).

This simplifies to \(\frac{x \sec^2 x - \tan x}{x^2}\).