This is the first paper from a series of three in which the author introduces and studies a new class of subdifferentials for arbitrary functions. The paper is devoted to the finite dimensional case. Let X be finite dimensional, f:\(X\to \bar R\) a function and f(x)\(\in R\). Define \(d^-f(x;h)=\lim_{u\to h,}\inf_{t\searrow 0}(f(x+tu)-f(x))/t, \partial^-f(x)=\{x^*\in X^*:\quad<x^*,h>\leq d^-f(x;h)\quad for\quad all\quad h\}\) and \(\partial_ af(z)=\limsup \partial^-f(x)\) for \(x\to z\), f(x)\(\to f(z)\). For this kind of subdifferential, which is always included in Clarke’s subdifferential, and generally is not convex, geometric characterizations, formulas (in terms of inclusions) for the subdifferential of the sum, maximum and minimum of two functions, and also for composition are given. Surjection and stability results and necessary conditions for optimality for minimization problems are stated in terms of approximate subdifferentials. At the end of the paper it is shown that the approximate subdifferential is minimal among all the subdifferentials which satisfy some natural conditions.