Find the derivative of $LaTeX: \displaystyle f(x) = \csc^{-1}{\left(9 x \right)}$ .

Using the chain rule gives $LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dx}$ where $LaTeX: \displaystyle f(u) = \csc^{-1}{\left(u \right)}$ and $LaTeX: \displaystyle u = 9 x$ gives $LaTeX: \displaystyle f'(x) = - \frac{1}{9 x^{2} \sqrt{1 - \frac{1}{81 x^{2}}}}$