Differentiation on constant value? let $f, g, h$ be functions with same domains and same ranges. and if I use chain rule to compute $$\frac{df}{dg}=\frac{df}{dh}\frac{dh}{dg},$$
what if $h$ is constant? I mean, $\Delta h=0$? 
I guess then $f$ should be constant, but I can't tell why.

The notation is very convenient, but a little rigorously deceptive. What is really happening here is that you have several compositions happening subtly. What your chain rule actually means is:
$$
\underbrace{(f\circ h)' \circ g}_{\frac{df}{dg}} = \big[\,\underbrace{f' \circ (h\circ g)}_{\frac{df}{dh}}\big]\cdot \big[ \underbrace{h' \circ g}_{\frac{dh}{dg}}\big]
$$
In particular, if $h$ is constant, then $f\circ h$ is also constant, so $(f\circ h)' \equiv 0$. 
