In this paper, we introduce a higher order differential cryptanalysis framework based on higher order derivatives. We show the previous attacks, higher order differential attack(HODA), algebraic IV differential attack(AIDA), Cube Attack, Cube Tester and bitwise higher order differential attack(BHODA) are all theoretically based on higher order derivatives and can be generalized in the cryptanalysis framework. Inspiring from the framework, we show some properties of higher order derivatives and discuss their applications. We also propose a practical higher order differential cryptanalysis technique and an algorithm based on it, additionally.