The study of derivatives of cubic Boolean functions is essential to tackle some related problems, such as the existence of cubic APN permutations in even dimension (the cubic case of the so-called "big APN problem"). We will show new numerical relations involving balancedness and weights of derivatives, of order one and two, as well as a conjecture that might reveal an unexpected behaviour. These are joint results with Augustine Musukwa (Univ. of Mzuzu, Malawi).