The standard definition of differentiability of a function f of one real variable x given on an open domain U with values in the real numbers uses the limit of the difference quotient. But by a simple formula manipulation, one finds that this is equivalent to the following statement: Such a function f is differentiable at a place c in U if there is a real number d and a function Δ defined on U that tends to 0 as x tends to c such that

f(x) = f(c) + d · (x – c) + (x – c) · Δ(x)

holds for each x in U. Put it differently: A function f is differentiable at c if and only if it behaves near to c like an affine-linear function plus a (convenient) error term.

As a characterization of differentiability, this statement can already be found in several textbooks from the 19^th and beginning 20^th century. But it was Karl Weierstraß (1815–1897) who used it as the definition of differentiability in his 1861 lecture course on differential calculus at the “Königliches Gewerbe-Institut” in Berlin in the case of one variable. Later on, Constantin Carathéodory (1873–1950) advocated this version of the definition, which made it known in particular in applied analysis in Germany. Still in 1998, Donald E. Knuth (*1938) propagated this approach to differentiability, in combination with Edmund Landau’s (1877–1938) “O” notation.

Admittedly, this definition of differentiability has the disadvantage that one does not immediately see how to calculate the derivative of f at the place c in contrast to dealing with the limit of the difference quotient (f(x) – f(c)) / (x – c). Therefore, one may argue that it is not advisable to use this definition as the primary one in secondary school mathematics teaching.

But on the other hand, is has several advantages for a mathematics course at universities and other institutions of tertiary education:

The first one, which has already been emphasized by several authors throughout the decades, is that one can avoid the technical complications in the proof of the chain rule for a composition f ο g of two functions f and g that arise from the possible vanishing of the denominator g(x) – g(c) appearing in the standard proof. (The proofs of the rules for the derivatives of sums, products, and quotients of functions are comparable with respect to the degree of technicality.)

The second advantage especially comes into bearing if one teaches a course that on the one hand treats not only one but also several variables and on the other hand is at least linked to a course in linear algebra: The map ξ → d·ξ is a linear map L so that–as already mentioned–the formula given above reads as

f(x) = f(c) + L(x – c) + (x – c) ·Δ(x) .

Furthermore, one can substitute the factor (x – c) in front of Δ(x) by its absolute value |x – c| which is nothing but a real number. (The slight change of Δ caused by this goes without saying.) This version of the definition can now be easily generalized to functions with vectors both as arguments and as values. Additionally, the same formulas can be used for complex valued functions of complex arguments.

The author of the contribution has given a course in analysis for an audience consisting both of teacher’s students and students of computer science at the University of Koblenz for several times. Here he first reminded the students of the definition of differentiability by the limit of the difference quotient with which they had been confronted at secondary school. But then he consequently turned to the other characterization which enabled him to get a smooth advance from one to several variables.