(a) Differentiate (sin x)/(√(cos x)) with respect to x.

Let $\displaystyle y=\frac{\sin \mathrm{x}}{\sqrt{\cos \mathrm{x}}}$

\[\begin{aligned} \frac{d y}{d \mathrm{x}} & =\frac{\sqrt{\cos \mathrm{x}} \cdot \cos \mathrm{x}-\sin \mathrm{x} \cdot\left(\dfrac{-\sin \mathrm{x}}{2 \sqrt{\cos \mathrm{x}}}\right)}{\cos \mathrm{x}} \\ \Rightarrow \frac{d y}{d \mathrm{x}} & =\frac{2 \cos ^{2} \mathrm{x}+\sin ^{2} \mathrm{x}}{2(\cos \mathrm{x})^{3 / 2}} \text { or } \frac{1+\cos ^{2} \mathrm{x}}{2(\cos \mathrm{x})^{3 / 2}} \end{aligned} \]

$\displaystyle y=5 \cos \mathrm{x}-3 \sin \mathrm{x}$, then $\displaystyle \frac{d y}{d \mathrm{x}}=-5 \cdot \sin \mathrm{x}-3 \cdot \cos \mathrm{x}$

\[\begin{aligned} \Rightarrow \frac{d^{2} y}{d \mathrm{x}^{2}}= & -5 \cdot \cos \mathrm{x}+3 \cdot \sin \mathrm{x}=-y \\ & \Rightarrow \frac{d^{2} y}{d \mathrm{x}^{2}}+y=0 \end{aligned} \]