  The ordinary generating function of the number of complete subgraphs
(cliques) of $G$, denoted by $C(G,x)$, is called the The clique polynomial of
the graph $G$. In this paper, we first introduce some \emph{clique} incidence
matrices associated by a simple graph $G$ as a generalization of the classical
vertex-edge incidence matrix of $G$. Then, using these clique incidence
matrices, we obtain two clique-counting identities that can be used for
deriving two combinatorial formulas for the first and the second derivatives of
clique polynomials. Finally, we conclude the paper with several open questions
and conjectures about possible extensions of our main results for higher
derivatives of clique polynomials.
