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