3.8: Trigonometric small change and second derivative

Variables $x$ and $y$ are such that $y = \dfrac{1 + \cos^{2} x}{\tan x}$. Use differentiation to find the approximate change in $y$ as $x$ increases from $\dfrac{\pi}{4}$ to $\dfrac{\pi}{4} + h$, where $h$ is small.

Given that $y = (3 - x)^{\frac{1}{3}}$, show that $y\dfrac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} - \left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^{2} = -\dfrac{1}{3}(3 - x)^{-\frac{4}{3}}$.