  In this paper, we study the derivatives of an integer-valued polynomial of a
given degree. Denoting by $E_n$ the set of the integer-valued polynomials with
degree $\leq n$, we show that the smallest positive integer $c_n$ satisfying
the property: $\forall P \in E_n, c_n P' \in E_n$ is $c_n = \mathrm{lcm}(1 , 2
, \dots , n)$. As an application, we deduce an easy proof of the well-known
inequality $\mathrm{lcm}(1 , 2 , \dots , n) \geq 2^{n - 1}$ ($\forall n \geq
1$). In the second part of the paper, we generalize our result for the
derivative of a given order $k$ and then we give two divisibility properties
for the obtained numbers $c_{n , k}$ (generalizing the $c_n$'s). Leaning on
this study, we conclude the paper by determining, for a given natural number
$n$, the smallest positive integer $\lambda_n$ satisfying the property:
$\forall P \in E_n$, $\forall k \in \mathbb{N}$: $\lambda_n P^{(k)} \in E_n$.
In particular, we show that: $\lambda_n = \prod_{p \text{ prime}}
p^{\lfloor\frac{n}{p}\rfloor}$ ($\forall n \in \mathbb{N}$).
