Math 2.0

statement theorem

Derivative of x^a

Statement

Let aa be a real number. The function x↦xax \mapsto x^a is differentiable on (0,∞)(0, \infty), and ddxxa=axa−1for x>0.\frac{d}{dx} x^a = a x^{a-1} \quad \text{for } x > 0.

Remarks

Write xa=ealn⁡xx^a = e^{a \ln x}. The chain rule with exp⁡′=exp⁡\exp' = \exp and ln⁡′x=1/x\ln' x = 1/x gives ealn⁡x⋅ax=axa−1e^{a \ln x} \cdot \frac{a}{x} = a x^{a-1}. For example, ddxx=12x\frac{d}{dx} \sqrt{x} = \frac{1}{2 \sqrt{x}}. Similarly, ddxax=axln⁡a\frac{d}{dx} a^x = a^x \ln a for a fixed a>0a > 0. Mathlib states the main claim as Real.hasDerivAt_rpow_const.

Explanations

L1 human

The power rule for whole exponents also works for real exponents, if x>0x > 0. Bring the exponent down and lower it by one.

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To read this node, read these 60 nodes in this order. The foundations come first, and the list ends with this node.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Strictly monotone function definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Real sequence definition
  8. Subsequence definition
  9. Continuity at a point definition
  10. Derivative definition
  11. One-sided limit definition
  12. Bounded sequence definition
  13. Geometric sequence definition
  14. Limit of a sequence definition
  15. Monotone sequence definition
  16. Series definition
  17. Continuous function definition
  18. Epsilon-delta definition of continuity statement
  19. Fermat's theorem on stationary points statement
  20. Sum of a finite geometric series statement
  21. Every sequence has a monotone subsequence statement
  22. Monotone convergence theorem for sequences statement
  23. Convergent sequence definition
  24. Squeeze theorem statement
  25. Cauchy product definition
  26. Partial sum definition
  27. Power series definition
  28. Intermediate value theorem statement
  29. Composition of continuous functions is continuous statement
  30. One-sided limits of a monotone function exist statement
  31. Bolzano–Weierstrass theorem statement
  32. The sequence r^n tends to zero for |r| < 1 example
  33. Radius of convergence definition
  34. Convergent series definition
  35. Chain rule statement
  36. Boundedness theorem statement
  37. Continuity of the inverse of a strictly monotone continuous function statement
  38. Absolute convergence definition
  39. A series of nonnegative terms converges iff its partial sums are bounded statement
  40. Sum of a series definition
  41. Extreme value theorem statement
  42. Derivative of an inverse function statement
  43. Comparison test statement
  44. Geometric series example
  45. Cauchy product of absolutely convergent series statement
  46. Rolle's theorem statement
  47. Ratio test statement
  48. Mean value theorem statement
  49. Ratio formula for the radius of convergence statement
  50. A function with positive derivative is strictly increasing statement
  51. The exponential series has infinite radius of convergence statement
  52. Exponential function definition
  53. Exponential of a sum statement
  54. The exponential function is positive statement
  55. The exponential function is its own derivative statement
  56. The exponential function is strictly increasing statement
  57. Natural logarithm definition
  58. Real exponentiation definition
  59. Derivative of the logarithm statement
  60. Derivative of x^a statement

Sources

  1. Web https://en.wikipedia.org/wiki/Power_rule, Proof for real exponents

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