Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE 3.1

Find $\frac{dy}{dx}$ when $y=\frac{e^x}{1+x^2}$

Find $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ when: $y={\tan }^{-1}(\sec x-\tan x)$.  [Write the answer in fraction.]

Find $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ when: $y={\tan }^{-1}\sqrt{\frac{1+\sin x}{1-\sin x}},\left(0<x<\frac{\mathrm{\pi }}{2}\right)$   [Write the answer in fraction.]

Find $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ when: $y={\tan }^{-1}\left(\frac{\sin x}{1+\cos x}\right)$  {Write the answer in fraction.}

Find $\frac{dy}{dx}$ when: $y={\tan }^{-1}\sqrt{\frac{1+\cos 2x}{1-\cos 2x}},\left(0<x<\frac{\mathrm{\pi }}{2}\right)$

Find $\frac{dy}{dx}$ when: $y={\cos }^{-1}\left(\frac{2\cos x-3\sin x}{\sqrt{13}}\right)$

Find $\frac{dy}{dx}$ when: $y={\sin }^{-1}\left(\frac{\cos x-2\sin x}{\sqrt{5}}\right)$

If $\mathrm{y}={\tan }^{-1}\frac{3\mathrm{x}}{1+4{\mathrm{x}}^2}+{\tan }^{-1}\frac{2+5\mathrm{x}}{5-2\mathrm{x}}$, then show that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{4}{1+16{\mathrm{x}}^2}$.