The aim of the paper is to extend Taylor’s formula to

${C}^{k,1}$ functions, i.e., functions whose

$k$th order derivatives are locally Lipschitz. First, the author defines the

$(k+1)$th order subdifferential of a

${C}^{k,1}$ function and gives a chain rule for this subdifferential. Then, two versions of Taylor’s theorem are established. A calculus rule for generalized Hessian of implicit functions is also presented. The results are then applied to derive high-order optimality conditions and second-order characterizations of quasiconvex functions.