Math 2.0

Real functions Proposition

Power rule

Builds on DerivativeProduct rule

Statement

Let n≥1n \ge 1 be a natural number. Then the function x↦xnx \mapsto x^n is differentiable on ℝ\mathbb{R}, and its derivative at xx is nxn−1n x^{n-1}.

Remarks

Induction on nn with the product rule gives (xn+1)′=(xn⋅x)′=nxn−1⋅x+xn=(n+1)xn(x^{n+1})' = (x^n \cdot x)' = n x^{n-1} \cdot x + x^n = (n + 1) x^n. The same formula holds for negative integers nn when x≠0x \ne 0, and for real exponents when x>0x > 0. Mathlib states it as hasDerivAt_pow.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementProduct rule, statement, no evidence levelProduct rulestatementPower rule, statement, no evidence levelPower rulestatement
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. Limit of a function definition
  3. Continuity at a point definition
  4. Derivative definition
  5. Differentiable function definition
  6. Differentiable functions are continuous statement
  7. Product rule statement
  8. Power rule statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Power_rule, Proof by induction (natural numbers)
About this page
Tag0JZGW75C5Z
URIm2://0JZGW75C5Z
Kindstatement, proposition
WritingW1 stated
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0JZGW75C5Z.
UsesDerivative, Product rule
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