Math 2.0

statement corollary

Derivative of a polynomial

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.

Explanations

L1 human

We differentiate a polynomial term by term with the power rule. For example, the derivative of 3x2+5x+73 x^2 + 5 x + 7 is 6x+56 x + 5.

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To read this node, read these 15 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Polynomial, Calculus

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