\[
\begin{aligned}
y &= \left(\sin x \right)^{\tan x} \\[2.0ex]
\ln y &= \tan x \cdot \ln (\sin x) \\[2.0ex]
\frac{1}{y} \frac{dy}{dx} &= \left(\frac{d}{dx} (\tan x) \right) \cdot \ln (\sin x) + \tan x \cdot \left(\frac{d}{dx} (\ln (\sin x)) \right) \\[2.0ex]
\frac{1}{y} \frac{dy}{dx} &= \sec^2 x \cdot \ln (\sin x) + \tan x \cdot \frac{\cos x}{\sin x} \\[2.0ex]
\frac{1}{y} \frac{dy}{dx} &= \sec^2 x \cdot \ln (\sin x) + 1 \\[2.0ex]
\frac{dy}{dx} &= y \cdot \left(\sec^2 x \cdot \ln (\sin x) + 1 \right) \\[2.0ex]
\frac{dy}{dx} &= (\sin x)^{\tan x} \cdot \left(\sec^2 x \cdot \ln (\sin x) + 1 \right)
\end{aligned}
\]
