Let $R$ be an arbitrary commutative ring. If $$f(X) \,:=\, \sum_{i=1}^na_iX^i$$ is a polynomial in the ring $R[X]$ , one can form in a polynomial ring $R[X,\,Y]$ the polynomial $$f(X\!+\!Y) \,=\, \sum_{i=1}^na_i(X\!+\!Y)^i.$$ Expanding this by the powers of $Y$ yields uniquely the form

(1)

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

It is apparent that this algebraic definition of derivative of polynomial is in harmony with the definition of derivative 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 $$(fg)' = f'g+g'f$$ with its generalisations. Especially: $$(X^n)' = nX^{n-1} \quad\mbox{for}\;\; n = 1,\,2,\,3,\,\ldots$$

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 $R[[X]]$ 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 $f$ , denoting them by $\displaystyle\frac{\partial f}{\partial X_i}$ . Then, e.g. the ``Euler's theorem on homogeneous functions'' $$X_1\frac{\partial f}{\partial X_1}+X_2\frac{\partial f}{\partial X_2}+\ldots+X_m\frac{\partial f}{\partial X_m} \;=\; nf$$ is true for a homogeneous polynomial $f$ of degree $n$ .