  We generalize the Hahn variational calculus by studying problems of the
calculus of variations with higher-order derivatives. The symmetric quantum
calculus is studied, namely the $\alpha,\beta$-symmetric, the $q$-symmetric,
and the Hahn symmetric quantum calculus. We introduce the symmetric quantum
variational calculus and an Euler-Lagrange type equation for the $q$-symmetric
and Hahn's symmetric quantum calculus is proved. We define a symmetric
derivative on time scales and derive some of its properties. Finally, we
introduce and study the diamond integral, which is a refined version of the
diamond-$\alpha$ integral on time scales.
