In this paper, we choose the derivative polynomials for tangent and secant as basis sets of polynomial space. From this viewpoint, we first give an expansion of the derivative polynomials for tangent in terms of the derivative polynomials for secant, and we then present a result in the reverse direction. We also discuss the relationships between alternating derivative polynomials and Eulerian polynomials. As applications, we give certain expansions of the alternating derivative polynomials, which indicate that the alternating derivative polynomials share more properties with the Chebyshev polynomials.