Formal Derivatives

Formal Derivatives, An Introduction

Introduction

Let R be any commutative ring, and let p(x) be a polynomial with coefficients in R. The formal derivative of p(x) is produced by folding the exponents into the coefficients, and decreasing the exponents by 1. Thus 4x³+5x+1 becomes 12x²+5.

If you've studied differential calculus, this definition will look familiar. But it has nothing to do with calculus; hence it is called a formal derivative. For instance, R might have characteristic n. In other words, R is an extension of the integers mod n. In such a ring, the derivative of xⁿ-1 is 0, simply because nx^n-1 is 0.

Verify that the derivative of p(x)+q(x) is the sum p′+q′, and the derivative of c×p(x) is c times p′. This follows directly from the definition.

Next prove the product rule for differentiation. Expand p*q using the distributive property, giving a sum of various terms c×xⁱ×x^j. The derivative of such a term is c×(i+j)×x^i+j-1. This is the same as,

c × (i×x^i-1×x^j + j×x^j-1×xⁱ)

Regroup terms to show the derivative of p×q is p′×q + p×q′.

Using induction on n, prove that the derivative of p(x)ⁿ is n times p(x)^n-1 times p′(x). Spread this across an entire polynomial and we have the chain rule. The derivative of q(p) is q′(p)×p′.