  The cases when the equation for the derivative of the confluent Heun function
has only three singularities (in general, the equation has four such points)
are examined. It is shown that this occurs only in three specific cases.
Further, it is shown that in all these three cases this equation is reduced to
some confluent Heun equation with changed parameters. This means that in these
cases the derivative of the confluent Heun function is expressed via some other
confluent Heun function. The analysis of the obtained relations shows that they
can be useful for the construction of some exact solutions of the confluent
Heun equation for specific values of involved parameters. Several examples of
such cases are presented; and for all the cases the final solutions in terms of
simpler special functions are explicitly presented. In addition, it is shown
that in the mentioned three cases, the solution can be expanded as a series in
terms of generalized hypergeometric functions of Goursat or Clausen.
