Let be an arbitrary commutative ring. If
is a polynomial in the ring , one can form in a polynomial ring the polynomial
Expanding this by the powers of yields uniquely the form
where and .
We define the polynomial in (1) the derivative of the polynomial and denote it by or
It is apparent that this algebraic definition of derivative of
polynomial is in harmony with the definition of
derivative of analysis when is
or ; then we identify the
substitution homomorphism and
the polynomial function.
It is easily shown the linearity of the derivative of polynomial and the product rule
with its generalisations. Especially:
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 , we can analogically define the partial derivatives of , denoting them by . Then, e.g. the “Euler’s theorem on homogeneous functions”
is true for a homogeneous polynomial of degree .