We generalize the fractional Caputo derivative to the fractional derivative
${{^CD}^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left
Caputo fractional derivative of order $\alpha$ and the right Caputo fractional
derivative of order $\beta$. The fractional variational problems under our
consideration are formulated in terms of ${{^CD}^{\alpha,\beta}_{\gamma}}$. The
Euler-Lagrange equations for the basic and isoperimetric problems, as well as
transversality conditions, are proved.