Найти производную функции $f(x) = \sqrt{1+3\,\mathrm{tg}\,x}$.

$u = 1+3\,\mathrm{tg}\,x$, $u' = \dfrac{3}{\cos^2 x}$. $$ f'(x) = \frac{u'}{2\sqrt u} = \frac{3/\cos^2 x}{2\sqrt{1+3\,\mathrm{tg}\,x}} = \frac{3}{2\cos^2 x\sqrt{1+3\,\mathrm{tg}\,x}}. $$

$$ \boxed{f'(x) = \dfrac{3}{2\cos^2 x\,\sqrt{1+3\,\mathrm{tg}\,x}}} $$