While playing around with a few polynomials, I have come across something that seemed odd to me and I wanted to ask this community if they could clear me up on what happened here.
When I added up the inverses of the derivatives at the roots of a polynomial, that sum turned out to be zero: $\sum_{r=0}^n \frac 1 {f'(x_r)} = 0$, where $f(x_r)=0$, $f'(x_r) \neq 0$ (No two roots are the same [Citation needed [Next question I can ask]]), $f(x)=\sum_{p=0}^na_n*x^n$ and $n \geq 2$ (Polynomial of degree 2 or greater)
Changing anything about the roots or the degree of the polynomial did not change the sum, except for when the polynomial had two or more identical roots, then the sum was (generally) not equal to zero
My question is: Is this hypothesis true, and if so, why is that, and why should it be true? Does it have other conditions other than having unlike roots? Please prove it, if possible, using only calculus, analysis and algebra, haven't learned about any of the more complex fields yet.
Thanks in advance!
