Question

$\mathrm{If}y=\left(\frac{2-3\cos x}{\sin x}\right),\mathrm{find}\frac{dy}{dx}\mathrm{at}x=\frac{\mathrm{\pi }}{4}$

Answer 1

$$\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}\left(\frac{2-3\cos x}{\sin x}\right) \\ =\frac{d}{dx}\left(\frac{2}{\sin x}\right)-\frac{d}{dx}\left(\frac{3\cos x}{\sin x}\right) \\ =2\frac{d}{dx}\left(\cos ecx\right)-3\frac{d}{dx}\left(\cot x\right) \\ =-2\cos ecx\cot x+3\cos e{c}^{2}x \\ \frac{dy}{dx}\text{at x=}\frac{\mathrm{\pi }}{4}\text{=}-2\cos ec\frac{\mathrm{\pi }}{4}\cot \frac{\mathrm{\pi }}{4}+3\cos e{c}^2\frac{\mathrm{\pi }}{4} \\ =-2\left(\sqrt{2}\right)\left(1\right)+3{\left(\sqrt{2}\right)}^2 \\ =-2\sqrt{2}+6 \\ =6-2\sqrt{2} \end{gathered}$$