The correct way seems to be
$$\frac{d}{dx} \cos^{-1}{(\ln{x})} = \frac{ \frac{-1}{x} }{\sqrt{1-(\ln{x})^2}} = - \frac{1}{x \sqrt{1-(\ln{x})^2}}$$
But why not
$\frac{d}{dx} \cos^{-1}{(\ln{x})} = \frac{d}{dx} (\cos{(\ln{x})})^{-1} = -1 (\cos{(\ln{x})})^{-2}(- \frac{1}{x} \sin{(\ln{x})}) = \frac{\sin{(\ln{x})}}{x \cdot \cos^2{(\ln{x})}}$
Is it wrong? Maybe its another careless mistake again?