derivative of polynomial

Let $R$ be an arbitrary commutative ring.  If

$$f(X):=\sum _{i=1}^n{a}_i{X}^i$$

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

$$f(X+Y)=\sum _{i=1}^n{a}_i{(X+Y)}^i.$$

Expanding this by the powers (http://planetmath.org/GeneralAssociativity) of $Y$ yields uniquely the form

$f(X+Y):=f(X)+f_1(X)Y+f_2(X,Y)Y^2,$ (1)

where  $f_1(X)\in R[X]$  and  $f_2(X,Y)\in R[X,Y]$.

We define the polynomial $f_1(X)$ in (1) the derivative of the polynomial $f(X)$ and denote it by $f^{\prime }(X)$ or ${\displaystyle \frac{df}{dX}}$.

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

$${(fg)}^{\prime }=f^{\prime }g+g^{\prime }f$$

with its generalisations.  Especially:

$${(X^n)}^{\prime }=n{X}^{n-1}\mathit{\hspace{1em}}\text{for}n=1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots }$$

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 (http://planetmath.org/FormalPowerSeries) $R[[X]]$ extend the concept of derivative of polynomial and obey laws.

If we have a polynomial  $f\in R[X_1,X_2,\mathrm{\dots },X_m]$,  we can analogically define the partial derivatives of $f$, denoting them by ${\displaystyle \frac{\partial f}{\partial X_i}}$.  Then, e.g. the “Euler’s theorem on homogeneous functions (http://planetmath.org/EulersTheoremOnHomogeneousFunctions)”

$$X_1\frac{\partial f}{\partial X_1}+X_2\frac{\partial f}{\partial X_2}+\mathrm{\dots }+X_m\frac{\partial f}{\partial X_m}=nf$$

is true for a homogeneous polynomial $f$ 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	Multiplicity
Related topic	DiscriminantOfAlgebraicNumber
Defines	derivative of the polynomial