derivative of polynomial
Let be an arbitrary commutative ring. If
Expanding this by the powers (http://planetmath.org/GeneralAssociativity) 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 (http://planetmath.org/Derivative2) of analysis when is or ; then we identify substitution homomorphism and 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 (http://planetmath.org/FormalPowerSeries) extend the concept of derivative of polynomial and obey 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 (http://planetmath.org/EulersTheoremOnHomogeneousFunctions)”
is true for a homogeneous polynomial of degree .
|Title||derivative of polynomial|
|Date of creation||2013-03-22 18:20:02|
|Last modified on||2013-03-22 18:20:02|
|Last modified by||pahio (2872)|
|Defines||derivative of the polynomial|