derivative of polynomial
Let R be an arbitrary commutative ring. If
f(X):=n∑i=1aiXi |
is a polynomial in the ring R[X], one can form in a polynomial ring R[X,Y] the polynomial
f(X+Y)=n∑i=1ai(X+Y)i. |
Expanding this by the powers (http://planetmath.org/GeneralAssociativity) 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 derivative 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 (http://planetmath.org/Derivative2) of analysis when R is ℝ or ℂ; then we identify substitution homomorphism and polynomial function.
It is easily shown the linearity of the derivative of polynomial and the product rule
(fg)′=f′g+g′f |
with its generalisations. Especially:
(Xn)′=nXn-1 |
Remark. The polynomial ring may be thought to be a subring of , the ring of formal power series in . The derivatives defined in (http://planetmath.org/FormalPowerSeries) extend the concept of derivative of polynomial and obey laws.
If we have a polynomial , we can analogically define the partial derivatives of , denoting them by . Then, e.g. the “Euler’s theorem on homogeneous functions (http://planetmath.org/EulersTheoremOnHomogeneousFunctions)”
is true for a homogeneous polynomial of degree .
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 | Multiplicity![]() |
Related topic | DiscriminantOfAlgebraicNumber |
Defines | derivative of the polynomial |