Question

Prove: $\frac{d}{dx}(x+\frac{1}{x})=1-\frac{1}{x^2}$

Answer 1

$\frac{d}{dx}(x+\frac{1}{x})$

$\Rightarrow \frac{d}{dx}(f+g)=\frac{d}{dx}\left(f\right)+\frac{d}{dx}\left(g\right)$

$\Rightarrow \frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(\frac{1}{x}\right)$

$\Rightarrow 1+\frac{d}{dx}\left(x^{-1}\right)$

$\Rightarrow 1+...\dots -1.x^{-1-1}$

$\Rightarrow 1-\left(1.x^{-2}\right)$

$\Rightarrow 1-\frac{1}{x^2}$.