# 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

 $\displaystyle f(X\!+\!Y)\,:=\,f(X)+f_{1}(X)\,Y+f_{2}(X,\,Y)\,Y^{2},$ (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 $\mathbb{R}$ or $\mathbb{C}$; 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:

 $(X^{n})^{\prime}=nX^{n-1}\quad\mbox{for}\;\;n=1,\,2,\,3,\,\ldots$

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  $f\in R\,[X_{1},\,X_{2},\,\ldots,\,X_{m}]$,  we can analogically define the 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)”

 $X_{1}\frac{\partial f}{\partial X_{1}}+X_{2}\frac{\partial f}{\partial X_{2}}+% \ldots+X_{m}\frac{\partial f}{\partial X_{m}}\;=\;nf$

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