The aim of this paper is to find an expression for second derivative of the
function $\phi(t)$ defined by $$\phi(t) = \lt(\int_V \vphi(t,x)^{-\beta}
dx\rt)^{-\frac1{\be -n}},\qquad \beta\not= n,$$ where $U\subset \R$ and
$V\subset \R^n$ are open bounded subsets, and $\vphi: U\times V\to \R_+$ is a
$C^2-$smooth function. As a consequence, this result gives us a direct proof of
the dimensional Pr\'ekopa's theorem based on a local approach.Comment: 9 pages, to appear in J. Math. Anal. App