  We give a proof and extension of two formulas of Frobenius and Stickelberger
of Differential Calculus that they used in a fundamental paper concerning
elliptic functions theory. Our main ingredient is the introduction of a
bilinear form which is a vast generalization of Leibniz formula. This bilinear
form is also connected to the Lagrange product formula. As a corollary, we give
a different proof of a theorem of P.J. Olver.
