  The derivative polynomials introduced by Knuth and Buckholtz in their
calculations of the tangent and secant numbers are extended to a multivariable
$q$--environment. The $n$-th $q$-derivatives of the classical $q$-tangent and
$q$-secant are each given two polynomial expressions.
  The first polynomial expression is indexed by triples of integers, the second
by compositions of integers.
  The functional relation between those two classes is fully given by means of
combinatorial techniques. Moreover, those polynomials are proved to be
generating functions for so-called $t$-permutations by multivariable
statistics. By giving special values to those polynomials we recover classical
$q$-polynomials such as the Carlitz $q$-Eulerian polynomials and the
$(t,q)$-tangent and -secant analogs recently introduced. They also provide
$q$-analogs for the Springer numbers. Finally, the $t$-compositions used in
this paper furnish a combinatorial interpretation to one of the Fibonacci
triangles.
