Let $ L_{d}^{(1)}(x)$ denote the generalized Laguerre polynomial of degree $d$ and order $\alpha=1$. Clearly, since all the roots $r_1,\dots,r_d$ of $L_{d}^{(1)}$ are simple, there exists a strictly positive function $C=C(d)$ for which $$ \min_{i=1,\dots,d} |L'^{(1)}_{d} (r_i)| \geq C(d) > 0.$$

I would like to have a lower asymptotic bound for $C(d)$ (i.e. as rapidly increasing/slowly decreasing as possible).