This paper is concerned with the construction of quadrature rules that include approximations to derivative correction terms for standard formulas. The correction terms require only the use of the integrand values from the original formula. This is a standard method for improving simple quadrature formulas. What is new in this paper is that the quadrature formulas that are considered do not require the integrand values to be determined at equally spaced points. The main theoretical contributions of the paper are the derivation of some recursive formulas for the correction terms. The application of the new formulas is considered for both smooth and singular integrands. Two numerical examples are used to illustrate the new formulas.