Repeated derivatives of composite functions and generalizations of the Leibniz rule

We use the properties of Hermite and Kampé de Fériet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. These results are extended to product of functions of the above argument, thus giving rise to expressions which can formally be interpreted as generalizations of the familiar Leibniz rule. Finally, examples of practical interest are discussed.