We prove in the general setting of lower semicontinuous functions on Banach spaces the relation between the Rockafellar directional derivative and the mixed lower limit of the lower Dini derivatives. As a byproduct we derive the famous inclusions of tangent cones of closed sets in Banach spaces. The results are established using as principal tool multidirectional mean value inequalities [Aussel et al., SIAM J Optim 9(3), 690-706 (1999)]. © 2007 Springer-Verlag.
Correa, R., Gajardo, P., & Thibault, L. (2009). Links between directional derivatives through multidirectional mean value inequalities. Mathematical Programming, 57-77. https://doi.org/10.1007/s10107-007-0131-7