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We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to getclosed forms for the repeated derivatives of functions whose argument is aquadratic or higher-order polynomial. The results we obtain are extended toproduct of functions of the above argument, thus giving rise to expressionswhich can formally be interpreted as generalizations of the familiar Leibnizrule. Finally, examples of practical interest are discussed.

Repeated derivatives of composite functions and generalizations of the Leibniz rule