If $P(x)$ is a polynomial of degree $n > 1$ with only simple roots $a\_1,\ldots,a\_n$, is it true that $\frac 1{P'(a\_1)} + \cdots + \frac 1{P'(a\_n)} = 0$, and, if so, what is the proof? I can see this directly for $n = 2,3,4$ with some brute force for $4$.