m-th derivative polynomial zero
-th derivative of a polynomial function at
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 -th derivatives at of a polynomial function and is given.
A polynomial function can be presented in a form of
First derivative of is simply
second derivative is
third derivative is
and fourth derivative is
which can be later generalized until -th derivative as
For it can be obtained that
since the other terms are zero due to existence of with .
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]. ↩︎
AI, “Derivative of Polynomial”, StudySmarter, 4 Apr 2023, url https://www.studysmarter.co.uk/explanations/engineering/engineering-mathematics/derivative-of-polynomial/ [20241007]. ↩︎
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]. ↩︎