Perhaps the most powerful, yet elementary, tool in the analysis of functions of a single variable is the notion of derivative. It may be argued that it has both a very intuitive geometric flavour, the idea of line tangent to a curve at a point, and a just as natural interpretation as measure of growth. The many far-reaching applications of this one-variable concept strongly suggest that analogous concepts should be available for functions of several variables, and certainly indicate that such analogues are highly desirable. In the search of higher dimensional versions, however, one is immediately faced with the observation that while the idea of plane tangent to a surface at a point is a very direct, and possibly unique, way of extending the idea of line tangent to a curve, the idea of growth encounters basic ambiguities, for one has to first select a direction along which such growth is measured. The natural intuitive similarity between mathematical surfaces and physical terrain, the former being an abstraction of the latter, reveals indeed that when walking in mountaineous regions, upon selecting different directions one may find drastically different paths leading to the same point, and can often choose that with the desired degree of difficulty, namely of steepness. In one variable, on the contrary, there is essentially one and only one walking direction, hence a single possible slope. The analytic notions that arise are that of differentiability, which encodes the existence of the tangent (hyper)plane, and of directional derivative or partial derivative, which are weaker notions tailored to capture phenomena that are essentially one-dimensional.