Derivatives of Exponential, Logarithmic and Trigonometric Functions

Let \(f(x) = \frac{\cos{x}}{\sin{x}}\) for \(\sin{x} \neq 0\).

1. Use the quotient rule to show that \(f'(x) = \frac{-1}{\sin^2{x}}\).
2. Find \(f''(x)\).

In the following table, \(f'\left(\frac{\pi}{2}\right) = p\) and \(f' \left(\frac{\pi}{2} \right) = q\). The table also gives approximate values of \(f'(x)\) and \(f''(x)\) near \(x = \frac{\pi}{2}\).

3. Find the value of \(p\) and of \(q\).[3]
4. Use information from the table to explain why there is a point of inflexion on the graph of \(f\) where \(x = \frac{\pi}{2}\).[2]