  Given a fat point scheme $\mathbb{W}=m_1P_1+\cdots+m_sP_s$ in the projective
$n$-space $\mathbb{P}^n$ over a field $K$ of characteristic zero, the modules
of K\"ahler differential $k$-forms of its homogeneous coordinate ring contain
useful information about algebraic and geometric properties of $\mathbb{W}$
when $k\in\{1,\dots, n+1\}$. In this paper we determine the value of its
Hilbert polynomial explicitly for the case $k=n+1$, confirming an earlier
conjecture. More precisely this value is given by the multiplicity of the fat
point scheme $\mathbb{Y} = (m_1-1)P_1 + \cdots + (m_s-1)P_s$. For $n=2$, this
allows us to determine the Hilbert polynomials of the modules of K\"ahler
differential $k$-forms for $k=1,2,3$, and to produce a sharp bound for the
regularity index for $k=2$.
