HMMT 2002 K3

We are given the values of the differentiable real functions $f, g, h$, as well as the derivatives of their pairwise products, at $x=0$ :

$$f(0)=1 ; g(0)=2 ; h(0)=3 ; (g h)^{\prime}(0)=4 ; (h f)^{\prime}(0)=5 ; (f g)^{\prime}(0)=6 .$$

Find the value of $(f g h)^{\prime}(0)$.