Math 2.0

Special functions Theorem

The exponential function is its own derivative

Builds on Exponential functionExponential of a sumDerivative

Statement

The exponential function is differentiable on ℝ\mathbb{R}, and exp⁡′(x)=exp⁡(x)\exp'(x) = \exp(x) for all real xx.

Remarks

By the sum rule, exp⁡(x+h)−exp⁡(x)h=exp⁡(x)⋅exp⁡(h)−1h\frac{\exp(x + h) - \exp(x)}{h} = \exp(x) \cdot \frac{\exp(h) - 1}{h}. The series gives exp⁡(h)−1h→1\frac{\exp(h) - 1}{h} \to 1 as h→0h \to 0. Term-by-term differentiation of the power series gives the same result. The exponential function is the only solution of f′=ff' = f with f(0)=1f(0) = 1. Mathlib states the derivative as Real.hasDerivAt_exp.

Reading path

To understand this page from the ground up, read these 32 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 sequencedefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementCauchy product, definition, no evidence levelCauchy productdefinitionPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…exampleRadius of convergence, definition, no evidence levelRadius of convergencedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionSum of a series, definition, no evidence levelSum of a seriesdefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleCauchy product of absolutely convergent series, statement, no evidence levelCauchy product of abs…statementRatio test, statement, no evidence levelRatio teststatementRatio formula for the radius of convergence, statement, no evidence levelRatio formula for the…statementThe exponential series has infinite radius of convergence, statement, no evidence levelThe exponential serie…statementExponential function, definition, no evidence levelExponential functiondefinitionExponential of a sum, statement, no evidence levelExponential of a sumstatementThe exponential function is its own derivative, statement, no evidence levelThe exponential funct…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. Derivative definition
  6. Bounded sequence definition
  7. Geometric sequence definition
  8. Limit of a sequence definition
  9. Monotone sequence definition
  10. Series definition
  11. Sum of a finite geometric series statement
  12. Monotone convergence theorem for sequences statement
  13. Convergent sequence definition
  14. Squeeze theorem statement
  15. Cauchy product definition
  16. Partial sum definition
  17. Power series definition
  18. The sequence r^n tends to zero for |r| < 1 example
  19. Radius of convergence definition
  20. Convergent series definition
  21. A series of nonnegative terms converges iff its partial sums are bounded statement
  22. Absolute convergence definition
  23. Sum of a series definition
  24. Comparison test statement
  25. Geometric series example
  26. Cauchy product of absolutely convergent series statement
  27. Ratio test statement
  28. Ratio formula for the radius of convergence statement
  29. The exponential series has infinite radius of convergence statement
  30. Exponential function definition
  31. Exponential of a sum statement
  32. The exponential function is its own derivative statement

Where this leads

37 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Differential equation
About this page
Tag0EX9GFJ0B3
URIm2://0EX9GFJ0B3
Kindstatement, theorem
WritingW0 stub
MSC33B10, 26A24
StatusThis copy of the site has no build data. Run m2 status 0EX9GFJ0B3.
UsesExponential function, Exponential of a sum, Derivative

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