Question

Let $f\left(x\right)=x+\frac{1}{x}+\frac{1}{x}+\frac{1}{x}+......\infty$, find $f^{\prime }\left(x\right)$

Answer 1

$f\left(x\right)-x=\frac{1}{x}+\frac{1}{x}+\frac{1}{x}+.........$

$f^{\prime }\left(x\right)-1=\frac{1}{x^2}-\frac{1}{x^2}-\frac{1}{x^2}.....$

$f^{\prime }\left(x\right)-1=\frac{-1}{x}(\frac{1}{x}+\frac{1}{x}+\frac{1}{x}+......)$

$f^{\prime }\left(x\right)-1=\frac{-f\left(x\right)}{x}+1$

$f^{\prime }\left(x\right)-1=\frac{-f\left(x\right)}{x}+1$

$f^{\prime }\left(x\right)=2-\frac{f\left(x\right)}{x}$