(sin x)cos x
Let $y=(\sin x)^{\cos x}$
Taking log on both sides,
$\log y=\log (\sin x)^{\cos x}$
$\Rightarrow \log y=\cos x \cdot \log (\sin x)$ $\left[\because \log x^{y}=y \log x\right]$
Differentiating both sides w.r.t. $x$,
$\frac{1}{y} \cdot \frac{d y}{d x}=\frac{d}{d x} \cos x \cdot \log (\sin x)$
$\frac{1}{y} \cdot \frac{d y}{d x}=\cos x \cdot \frac{d}{d x} \log (\sin x)+\log (\sin x) \cdot \frac{d}{d x} \cos x$
$\frac{1}{y} \cdot \frac{d y}{d x}=\cos x \cdot \frac{1}{\sin x} \cdot \frac{d}{d x}(\sin x)+\log (\sin x) \cdot(-\sin x)$
$\Rightarrow \frac{1}{y} \cdot \frac{d y}{d x}=\cot x \cdot \cos x-\sin x \cdot \log (\sin x)$
$\frac{d y}{d x}=y[\cot x \cdot \cos x-\sin x \cdot \log (\sin x)]$
Thus, $\frac{d y}{d x}=(\sin x)^{\cos x}\left[\frac{\cos ^{2} x}{\sin x}-\sin x \cdot \log (\sin x)\right]$
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