Differentiate a product with constant-e exponent and a variable power

The exponent is the product $ex$, with $e$ constant. The product rule gives $e^{ex}[-\frac12\csc^2(x/2)]+\cot(x/2)e^{ex}e$. Factor to obtain $e^{ex}\left[e\cot(x/2)-\frac12\csc^2(x/2)\right]$. The cotangent requires $x\ne2k\pi$, $k\in\mathbb Z$.

For $x>0$, let $y=(2x)^{1/x}$ and take logs: $\ln y=x^{-1}\ln(2x)$. Thus $y\prime/y=x^{-1}(1/x)-x^{-2}\ln(2x)=[1-\ln(2x)]/x^2$. Therefore $y\prime=\dfrac{(2x)^{1/x}}{x^2}[1-\ln(2x)]$.