1.24: Quotient rule with trigonometric functions and rates of change

Find the exact value of $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ when $x = \dfrac{\pi}{3}$.

Hence find the approximate change in $y$ as $x$ increases from $\dfrac{\pi}{3}$ to $\dfrac{\pi}{3} + h$, where $h$ is small.

Given that $x$ is increasing at the rate of $3$ units per second, find the corresponding rate of change in $y$ when $x = \dfrac{\pi}{3}$, giving your answer in its simplest surd form.