Derivatives defined on!? I know Derivatives can be defined on  functions.My question is can we defined derivative on arbitrary relation?if not then whats the problem with it?(2) why we defined derivative only on real number ,i mean
what are requirement to define it on arbitrary topological vector space?

There are no derivatives but there are differentials defined e.g. the Gateaux differential and the Frechet differential . The Gateaux differential is defined on locally convex topological vector spaces:
$D(f,x_{0},v)=\displaystyle \lim_{\epsilon \to 0}\dfrac{f(x_{0}+\epsilon\,v)-f(x_{0})}{\epsilon}$
where $v$ is the direction of the differential.
The Frechet derivative is defined only in Normed Vector Spaces as follows:

As you can see the Frechet differential is independent of direction!
