real analysis - Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that

$\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$.

My thoughts: We have $P'(x) = P(x)(\frac1{x-x_1}+...+\frac1{x-x_n})$. So we may consider $P'(x)/P(x)$, which has poles at the roots of $P$.