derivative of polynomial

Let R be an arbitrary commutative ring.  If


is a polynomialMathworldPlanetmathPlanetmathPlanetmath in the ring R[X], one can form in a polynomial ringR[X,Y]  the polynomial


Expanding this by the powers ( of Y yields uniquely the form

f(X+Y):=f(X)+f1(X)Y+f2(X,Y)Y2, (1)

where  f1(X)R[X]  and  f2(X,Y)R[X,Y].

We define the polynomial f1(X) in (1) the derivativePlanetmathPlanetmath of the polynomial f(X) and denote it by f(X) or dfdX.

It is apparent that this algebraic definition of derivative of polynomial is in harmony with the definition of derivative ( of analysisMathworldPlanetmath when R is or ; then we identify substitution homomorphism and polynomial function.

It is easily shown the linearity of the derivative of polynomial and the product ruleMathworldPlanetmath


with its generalisations.  Especially:

(Xn)=nXn-1forn=1, 2, 3,

Remark.  The polynomial ring R[X] may be thought to be a subring of R[[X]], the ring of formal power series in X.  The derivatives defined in ( R[[X]] extend the concept of derivative of polynomial and obey laws.

If we have a polynomial  fR[X1,X2,,Xm],  we can analogically define the partial derivativesMathworldPlanetmath of f, denoting them by fXi.  Then, e.g. the “Euler’s theorem on homogeneous functions (”


is true for a homogeneous polynomialMathworldPlanetmathPlanetmath f of degree n.

Title derivative of polynomial
Canonical name DerivativeOfPolynomial
Date of creation 2013-03-22 18:20:02
Last modified on 2013-03-22 18:20:02
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Definition
Classification msc 13P05
Classification msc 11C08
Classification msc 12E05
Related topic DerivativesByPureAlgebra
Related topic PolynomialFunction
Related topic MultiplicityMathworldPlanetmath
Related topic DiscriminantOfAlgebraicNumber
Defines derivative of the polynomial