In numerous problems of applied statistical mechanics, such as the case of transport phenomena, the so-called derivatives under the integral have been an important and interesting tool to exploit, since in the study of these type of engineering systems, it is possible to perform a characterization of their behavior and know its general properties from their derivatives under the integral. Specifically, in the advanced mathematics, the treatment of the derivative under the integral is called the Leib- nitz’s rule available in some mathematical analysis books. In the present paper the demonstration of this theorem is achieved in the real case contributing to a better understanding from mathematicians, engineers and physicists.