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Question Number 129031 by oustmuchiya@gmail.com last updated on 12/Jan/21

$$Giventhat{\mathit{t}\mathit{a}\mathit{n}}^{-1}\mathit{x}showthat$$$$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\frac{1}{1+{\mathit{x}}^2}$$

Answered by MJS_new last updated on 12/Jan/21

$$y=\arctan x$$$$x=\tan y$$$$dx=\frac{dy}{{\cos }^{2}y}$$$$\frac{dy}{dx}={\cos }^{2}y$$$$\cos y=\frac{1}{\sqrt{1+{\tan }^{2}y}}\stackrel{y=\arctan x}{=}\frac{1}{\sqrt{1+x^2}}$$$$\Rightarrow$$$$\frac{dy}{dx}=\frac{1}{1+x^2}$$

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