derivatives by pure algebra

Let R be any commutative unique factorization domain (UFD) and x and h indeterminants. For instance, let R= the usual real numbers, or any other field. We treate R[x] as a subring of R[x,h].

We derive a definition for derivatives of polynomial and rational functions over R along with the usual rules: product ruleMathworldPlanetmath and power ruleMathworldPlanetmathPlanetmath. Despite the abstract nature of the definitions, the mechanics reflect the general understanding of introductory calculus, without any appeal to the Cauchy style ε-δ limits of analysisMathworldPlanetmath.

Definition 1.



where f(x+h)-f(x)=ha1(x,h)am(x,h) in the UFD R[x,h]. Furthermore, given g(x,h)R[x,h] define


(which is simply the evaluation homomorphism at h=0.) Finally define


We also denote dfdx by f(x).

Example 1.

First we reduce the fraction in a manner identical to the usual methods of calculus:

ddx(5x2-7x+9) = limh0(5(x+h)2-7(x+h)+9)-(5x2-7x+9)h
= limh05x2+10xh+5h2-7x-7h+9-5x2+7x-9h
= limh010xh+5h2-7hh
= limh010x+5h-7.

At this stage we must interpret the limh0. Because the limit notation simply means to evaluate this polynomialPlanetmathPlanetmath at h=0 we find:


This is in contrast to the typical approach where h is said to “approach” 0. However, no differencePlanetmathPlanetmath is found in the solution and almost no difference is found in the method, only in the interpretationMathworldPlanetmathPlanetmath of the method. ∎

Proposition 2.

The derivativePlanetmathPlanetmath formulaMathworldPlanetmathPlanetmath is well-defined. In particular, h divides f(x+h)-f(x)R[x,h] for every f(x)R[x], and the a1(x,h)am(x,h) are unique to f(x).


For all f(x),g(x)R[x], it follows


Furthermore, for all aR


So now if we take f(x)=a0+a1x++anxn, then h|(f(x+h)-f(x)) if h|((x+h)i-xi) for every i. When i=0, (x+h)0-x0=0 so h|((x+h)0-x0). Now take i>0 and use of the binomial theorem to find:

(x+h)i-xi = j=0i(ij)xi-jhj-xi
= j=1i(ij)xi-jhj
= hj=1i(ij)xi-jhj-1.

Hence h|(f(x+h)-f(x)).

As R is a UFD, so is R[x,h]. Also h is irreducible in R[x,h], and h|(f(x+h)-f(x)), so f(x+h)-f(x)=ha1(x,h)am(x,h) for some ai(x,h)R[x,h], 1im, with each ai(x,h) unique to f(x+h)-f(x) up to multiplicationPlanetmathPlanetmath by a unit of R[x,h], that is, a unit of R. In particular, a1(x,h)am(x,h) is unique to f(x+h)-f(x), and so unique to f(x). ∎

Remark 3.

Although potentially obtuse, the notation limh0 is a function, limh0:R[x,h]R[x] and has kernel Rh=(h). So we have R[x,h]/(h)R[x]. Therefore we may also write:


It is important that we always reduce the fractions so that we are not encountering any division by 0 at any stage.

Theorem 4.

Derivatives satisfy the following rules:

  • Linearity

    For f(x),g(x)R[x] and aR

  • Power Rule
  • Product Rule

This form of a formal derivative applies to any UFD and so it also applies to . Thus it is possible to express polynomial calculus in terms of algebraic theory without any proper use of limits. This obscures many of the geometric properties such as the slope of a tangent line to a graph. However, computationally this technique outlines how ε,δ-limits are not required for the computation of derivatives.

Although abstract algebra, such as quotients of rings, are required to properly understand R[x,h]/(h), this approach still provides elementary proofs of derivative rules like the product rule. Although it is not necessary, to draw a distinct between R[x] and R[x,h]/(h) one may use when we consider the expressions in R[x,h]/(h) if the distinction is clarifying.

1 Derivatives of rational functions

One may also generalize the derivative to apply to general rational function f(x)R(x) by observing 1=xnx-n. Therefore


Now solve for ddx(x-n).


Thus we also derive the usual quotient ruleMathworldPlanetmath:

Title derivatives by pure algebra
Canonical name DerivativesByPureAlgebra
Date of creation 2013-03-22 15:59:37
Last modified on 2013-03-22 15:59:37
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 16
Author Algeboy (12884)
Entry type Definition
Classification msc 26B05
Classification msc 46G05
Classification msc 26A24
Related topic DerivativeOfXn
Related topic AlternativeProofOfDerivativeOfXn
Related topic DerivativeOfPolynomial