Practise products, exponentials and a logarithmic square root

The product rule gives $\sin x+x\cos x$, matching the printed answer $x\cos x+\sin x$. All real $x$ are allowed, with angles in radians.

The chain rule gives $e^{1+\cos x}(-\sin x)=-\sin x\,e^{1+\cos x}$, matching the printed answer, for all real $x$.

Write $\ln\sqrt{1+\cos x}=\frac12\ln(1+\cos x)$. The logarithm requires $1+\cos x>0$, excluding odd multiples of $\pi$. Differentiating gives $-\sin x/[2(1+\cos x)]$, exactly the printed answer. A zero radicand is not allowed inside this logarithm.