# 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
$\frac{df}{dX}$.

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 $\mathbb{R}$ or $\u2102$; 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 $\frac{\partial f}{\partial {X}_{i}}$. 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 $n$.

Title | derivative of polynomial |

Canonical name | DerivativeOfPolynomial |

Date of creation | 2013-03-22 18:20:02 |

Last modified on | 2013-03-22 18:20:02 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 12 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 13P05 |

Classification | msc 11C08 |

Classification | msc 12E05 |

Related topic | DerivativesByPureAlgebra |

Related topic | PolynomialFunction |

Related topic | Multiplicity^{} |

Related topic | DiscriminantOfAlgebraicNumber |

Defines | derivative of the polynomial |