The setting for the author's integral representations is the following: \(D\)~is a bounded domain in~\(\mathbb C^2\) with smooth boundary; the variety~\(V\) is the zero set of a function holomorphic in a neighborhood of the closure of~\(D\), and \(V\) is assumed to meet the boundary of \(D\) transversely in a smooth curve; \(M\)~is the intersection \(V\cap D\); and \(f\) is a function holomorphic in a neighborhood of the closure of~\(D\). The main result expresses derivatives of~\(f\) at points of~\(M\) in terms of a Cauchy-Fantappiè integral of~\(f\) over the boundary of~\(M\) and derivatives of functions defined by line integrals of~\(f\) over boundaries of varieties close to~\(M\). An application is given to the representation of analytic functionals on~\(\mathbb C^2\).