  In our previous publications we have introduced analogs of partial
derivatives on the algebras U(gl(N)). In the present paper we compare two
methods of introducing these analogs: via the so-called quantum doubles and by
means of a coalgebraic structure. In the case N=2 we extend the quantum partial
derivatives from U(u(2)) (the compact form of the algebra U(gl(2))) on a bigger
algebra, constructed in two steps. First, we define the derivatives on a
central extension of this algebra, then we prolongate them on some elements of
the corresponding skew-field by using the Cayley-Hamilton identities for
certain matrices with noncommutative entries. Eventual applications of this
differential calculus are discussed.
