New calculus rules are given for the Clarke generalized gradient \(\partial f\) of a general lower semicontinuous function \(f: {\mathbb{R}}^ n\to {\mathbb{R}}\cup \{\pm \infty \}\). These include rules for computing \(\partial f\) when \(f=f_ 1+f_ 2\circ F\) in cases where \(f_ 1\), \(f_ 2\) are l.s.c. and F is either stricly differentiable or isotone. Basing their analysis upon the generalized upper directional derivative \(f^{\uparrow}(x;v)\), the authors obtain the weakest conditions to date under which such formulas are valid; their method also readily yields conditions for equality in the formulae. Among the consequences of the new calculus rules are new necessary conditions for constrained mathematical programming problems, new formulae for the calculus of (Clarke) normal and tangent cones, and all the known results for the finite-dimensional subgradient calculus of convex functions. Other forms of generalized derivatives (contingent, Ursescu, etc.) are discussed, and there is a wealth of insightful commentary.