derivatives by pure algebra
We derive a definition for derivatives of polynomial and rational functions over along with the usual rules: product rule and power rule. Despite the abstract nature of the definitions, the mechanics reflect the general understanding of introductory calculus, without any appeal to the Cauchy style limits of analysis.
where in the UFD . Furthermore, given define
(which is simply the evaluation homomorphism at .) Finally define
We also denote by .
First we reduce the fraction in a manner identical to the usual methods of calculus:
At this stage we must interpret the . Because the limit notation simply means to evaluate this polynomial at we find:
This is in contrast to the typical approach where is said to “approach” . However, no difference is found in the solution and almost no difference is found in the method, only in the interpretation of the method. ∎
For all , it follows
Furthermore, for all
So now if we take , then if for every . When , so . Now take and use of the binomial theorem to find:
Derivatives satisfy the following rules:
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 -limits are not required for the computation of derivatives.
Although abstract algebra, such as quotients of rings, are required to properly understand , this approach still provides elementary proofs of derivative rules like the product rule. Although it is not necessary, to draw a distinct between and one may use when we consider the expressions in if the distinction is clarifying.
1 Derivatives of rational functions
One may also generalize the derivative to apply to general rational function by observing . Therefore
Now solve for .
Thus we also derive the usual quotient rule:
|Title||derivatives by pure algebra|
|Date of creation||2013-03-22 15:59:37|
|Last modified on||2013-03-22 15:59:37|
|Last modified by||Algeboy (12884)|