Let $f$ be an entire function of order $ρ<\infty$. Assume that $f$ does not vanish identically on $\mathbb{C}$. Then, we know that $f$ has a Hadamard's product formula


$$ f(s) =e^{g(s)}s^{r}\prod \_ {k=1}^{\infty}\frac{s \_ {k}-s}{s \_ {k}} e^{s/s \_ k} $$


the integer $r$ is the order of vanishing of $f$ at $s=0$, the $s\_{k}$ are the other zeros of $f$ listed with multiplicity, $g$ is a polynomial of degree at most $ρ$, and the product converges uniformly in bounded subsets of $ℂ$.
My question is how I can deduce directely a Hadamard's product formula for the derivative $f^′$ from the one of the function $f$. 



Hadamard's product formula for the derivative 
(1) It seems your formula for the Hadamard product is only correct for $\rho<2$; more generally the exponent of $e^{s/s\_k}$ contains a power series in $s$ of order $q={\rm Int}\;\rho$; see for example Eq. 1 in these [lecture notes](http://www.math.harvard.edu/~elkies/M259.06/prod.pdf).


(2) To find a similar expression for $f'$, just take the logarithmic derivative:


$$ f'(s)/f(s)=g'(s)+r/s+\sum\_{k=1}^{\infty}\frac{(s/s\_k)^q}{s-s\_k}.$$


