The concepts of the generalized derivative on direction, the subdifferential and the Clarke subdifferential have been introduced for locally Lipschitz functions. The locally Lipschitz function has been the difference of maximum functions from finite numbers of continuously differentiable functions. The main formulas for Clarke subdifferential calculus have been presented in the form of inclusions. The condition when the Clarke subdifferential has been equal to the difference of Clarke subdifferentials has been defined for locally Lipschitz functions.