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 $n$.