1. Formulas:

• $\frac{d}{dx}(\ln u) = \frac{1}{u} \cdot u'$
• $\frac{d}{dx}(\tan x) = \sec^2 x$
• Identidad: $\sin 2x = 2 \sin x \cos x$

2. Desarrollo:
$$ \begin{aligned} y' &= \frac{1}{\tan x} \cdot \sec^2 x \\ y' &= \frac{\cos x}{\sin x} \cdot \frac{1}{\cos^2 x} \\ y' &= \frac{1}{\sin x \cos x} \end{aligned} $$
Para simplificar, multiplicamos numerador y denominador por 2:
$$ y' = \frac{2}{2 \sin x \cos x} = \frac{2}{\sin 2x} $$

3. Resultado:
$$ \boxed{dy = \frac{2}{\sin 2x} dx} $$