notes

m-th derivative polynomial zero

Sparisoma Viridi
2 mins read ·

mm-th derivative of a polynomial function at x=0x=0

Using limit n-th derivative of a polynomial function can be easily found 1. Some theorems, such as power rule, sum & different rule, and constant multiple rule, are required in finding the derivatives 2. Using derivative critical and inflection points of a polynomial function can be studied 3.

Here a formula for mm-th derivatives at x=0x=0 of a polynomial function and is given.

A polynomial function can be presented in a form of

f(x)=i=0naixi.(1)\tag{1} f(x) = \sum_{i=0}^n a_i x^i.

First derivative of f(x)f(x) is simply

f(x)=i=1ni ai xi1,(2)\tag{2} f'(x) = \sum_{i=1}^n i \ a_i \ x^{i-1},

second derivative is

f(x)=i=2n(i1)i ai xi2,(3)\tag{3} f''(x) = \sum_{i=2}^n (i-1) i \ a_i \ x^{i-2},

third derivative is

f(x)=i=3n(i2)(i1)i ai xi3,(4)\tag{4} f'''(x) = \sum_{i=3}^n (i-2)(i-1) i \ a_i \ x^{i-3},

and fourth derivative is

fν(x)=i=4n(i3)(i2)(i1)i ai xi4,(5)\tag{5} f'^\nu(x) = \sum_{i=4}^n (i-3)(i-2)(i-1) i \ a_i \ x^{i-4},

which can be later generalized until mm-th derivative as

fm(x)=i=mn(im+1)(i3)(i2)(i1)i ai xim.(6)\tag{6} f^m(x) = \sum_{i=m}^n (i-m+1) \cdots (i-3)(i-2)(i-1) i \ a_i \ x^{i-m}.

For x=0x=0 it can be obtained that

fm(0)=m! am,(7)\tag{7} f^m(0) = m! \ a_m,

since the other terms are zero due to existence of xix^i with i>0i > 0.


  1. Thomas Wallace Colthurst, Craig B. Watkins, Joy Nicholson, Elizabeth Shapere, Carolyn Phillips, “Derivatives of Polynomials”, Worl Web Math, Massachusetts Institute of Technology, 28 Aug 1998, url https://web.mit.edu/wwmath/calculus/differentiation/polynomials.html [20241007]. ↩︎

  2. AI, “Derivative of Polynomial”, StudySmarter, 4 Apr 2023, url https://www.studysmarter.co.uk/explanations/engineering/engineering-mathematics/derivative-of-polynomial/ [20241007]. ↩︎

  3. Donald Byrd, “Polynomials and their Derivatives: Polynomials, Critical Points, and Inflection Points”, IUPUI Math Education, Indiana University Informatics, 30 Nov 2011, url https://homes.luddy.indiana.edu/donbyrd/Teach/Math/Polynomials+Derivatives.pdf [20241007]. ↩︎

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