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General Associativity Subring Polynomial Ring Ring Commutative Ring Derivative Formal Power Series Topic Entry On Analysis Partial Derivative Algebraic Element Polynomial Function Product Rule Homogeneous Polynomial Evaluation Homomorphism Eulers Theorem On Homogeneous Functions

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Derivative Of Polynomial

Let be an arbitrary commutative ring. If

$\displaystyle f(X) \,:=\, \sum_{i=1}^na_iX^i$

is a polynomial in the ring

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

$\displaystyle f(X\!+\!Y) \,=\, \sum_{i=1}^na_i(X\!+\!Y)^i.$

Expanding this by the powers of

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 in (1) the derivative of the polynomial and denote it by or $\displaystyle\frac{df}{dX}$ .

It is apparent that this algebraic definition of derivative of polynomial is in harmony with the definition of derivative of analysis when 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

$\displaystyle (fg)' = f'g+g'f$

with its generalisations. Especially:

$\displaystyle (X^n)' = nX^{n-1}$ for $\displaystyle \;\; n = 1,\,2,\,3,\,\ldots$

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

If we have a polynomial $f \in R\,[X_1,\,X_2,\,\ldots,\,X_m]$ , we can analogically define the partial derivatives of , denoting them by $\displaystyle\frac{\partial f}{\partial X_i}$ . Then, e.g. the “Euler's theorem on homogeneous functions”

$\displaystyle X_1\frac{\partial f}{\partial X_1}+X_2\frac{\partial f}{\partial X_2}+\ldots+X_m\frac{\partial f}{\partial X_m} \;=\; nf$

is true for a homogeneous polynomial

of degree

1	Polynomial Ring
1	Commutative Ring
1	Ring
1	Subring
1	General Associativity
2	Topic Entry On Analysis
2	Derivative
2	Formal Power Series
3	Partial Derivative
3	Algebraic Element
5	Product Rule
5	Polynomial Function
8	Homogeneous Polynomial
10	Evaluation Homomorphism
16	Eulers Theorem On Homogeneous Functions