calculus - Is this true for all polynomials

I took $3$ random polynomials with non zero roots one having even degree and two having odd degrees

1. $f(x)=\color{red}{4}x^2-(4\sqrt3+12)x+12\sqrt3$ having roots $\color{blue}{3,\sqrt3}$ and leading coefficient $\color{red}{4}$ and calculated values of $xf'(x)$$(f'(x)$ is the derivative of $f(x))$ at both roots which are $3f'(3)$ and $\sqrt3f'(\sqrt3)$ and then sum of their reciprocals $\frac1{3f'(3)}+\frac1{\sqrt3f'(\sqrt3)}=\frac{-1}{12\sqrt3}=\frac{-1}{\color{red}{4}}\left(\frac{1}{\color{blue}{3\cdot\sqrt3}}\right)$ then repeated same thing for


2. 
   

   $g(x)=\color{red}{1}x^3-\frac{20}{3}x^2-12x+\frac{32}{3}$ having roots $\color{blue}{8,-2,\frac{2}{3}}$

   
   

   $\frac1{8g'(8)}+\frac1{-2g'(-2)}+\frac1{\frac{2}{3}g'(\frac2 3)}=\frac{1}{\color{red}{1}}\left(\frac{1}{\color{blue}{8\cdot-2\cdot\frac2 3}}\right)$

   
   


3. 
   
   

   $h(x)=\color{red}{1}x^5+41x^4+137x^3-1601x^2-1818x+3240 $ having roots $\color{blue}{1,-2,5,-9,-36}$

   
   

   $\frac1{1h'(1)}+\frac1{-2h'(-2)}+\frac1{5h'(5)}+\frac1{-9h'(-9)}+\frac1{-36h'(-36)}=\frac{1}{\color{red}{1}}\left(\frac{1}{\color{blue}{1\cdot-2 \cdot5\cdot-9\cdot-36}}\right)$

   
   

Is this true for all polynomials? Is there any known result?