Differentiability of the "Positive Part" Map

This is a 1964 work which was quoted in some papers (in 1966, 1976) but remained unpublished. We hope the results and the hypotheses of the statements will still find some echo. We give here the English translation of the article "Un problème de dérivabilité dans $L^p$'', Séminaire d'Automatique Théorique, Université de Caen (France), November 27 (1964), only avoiding an archaical presentation of Uniform Integrability and suppressing two examples.