Math 2.0

Real functions Corollary

Derivative of a polynomial

Builds on Power ruleLinearity of differentiation

Statement

Let p(x)=∑k=0nakxkp(x) = \sum_{k=0}^{n} a_k x^k with real coefficients aka_k. Then pp is differentiable on ℝ\mathbb{R}, and p′(x)=∑k=1nkakxk−1.p'(x) = \sum_{k=1}^{n} k a_k x^{k-1}.

Remarks

It follows from the power rule and the linearity of differentiation. So the derivative of a polynomial of degree n≥1n \ge 1 is a polynomial of degree n−1n - 1. Mathlib states it as Polynomial.hasDerivAt.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementLimit of a sum of sequences, statement, no evidence levelLimit of a sum of seq…statementProduct rule, statement, no evidence levelProduct rulestatementLinearity of differentiation, statement, no evidence levelLinearity of differen…statementPower rule, statement, no evidence levelPower rulestatementDerivative of a polynomial, statement, no evidence levelDerivative of a polyn…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Derivative definition
  7. Limit of a sequence definition
  8. Differentiable function definition
  9. Epsilon-N definition of the limit of a sequence statement
  10. Differentiable functions are continuous statement
  11. Limit of a sum of sequences statement
  12. Product rule statement
  13. Linearity of differentiation statement
  14. Power rule statement
  15. Derivative of a polynomial statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Polynomial, Calculus
About this page
Tag0MH1WK89PP
URIm2://0MH1WK89PP
Kindstatement, corollary
WritingW0 stub
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0MH1WK89PP.
UsesPower rule, Linearity of differentiation

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