Find the derivative of LaTeX: \displaystyle f(x) = \cot^{-1}{\left(2 x \right)} .

Using the chain rule gives LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dx} where LaTeX: \displaystyle f(u) = \cot^{-1}{\left(u \right)} and LaTeX: \displaystyle u = 2 x gives LaTeX: \displaystyle f'(x) = - \frac{2}{4 x^{2} + 1}