Differentiation

Abstract

In the calculus of a function f of two real variables, i.e., of a two-dimensional vector variable (x,y), one usually works with the two partial derivatives ∂f/∂x and ∂f/∂y (to be formally defined in 3-4.1 below). The first of these is the limit of a certain quotient with numerator f(x+t,y)-f(x,y). In the terminology of vectors, this numerator may be written as f((x,y)+t(1,0))-f(x,y). If we now write simply x for (x,y) ∈ ℝ² and simply h for (1,0) ∈ ℝ², then the numerator can be expressed quite compactly as f(x+th)-f(x). With this notation, it becomes clearer that the partial derivative.