number e (Euler’s number)

• is a mathematical constant approximately equal to 2.71828…
• can be characterized in many ways:
  • it is the base of the natural logarithm
  • it is the summation of infinite series:
    • 𝑒 = 𝛴_{0≤𝑥≤∞}[1/𝑥!]
  • it is the limit of (1 + 1/𝑥)^𝑥 as 𝑥 approaches infinity (an expression that arises in the study of compound interest)
    • 𝑒 = 𝑙𝑖𝑚_𝑥→∞(1 + 1/𝑥)^𝑥
    • 𝑒^𝑎 = 𝑙𝑖𝑚_𝑥→∞(1 + 𝑎/𝑥)^𝑥
    • 𝑒^-𝑎 = 𝑙𝑖𝑚_𝑥→∞(1 - 𝑎/𝑥)^𝑥
  • it is the limit of (1 + 𝑥)^1/𝑥 as 𝑥 approaches 0
    • 𝑒 = 𝑙𝑖𝑚_𝑥→0(1 + 𝑥)^1/𝑥

  • it is also the unique positive number 𝑎 such that the exponential function 𝑓(𝑥) = 𝑎^𝑥 has a slope of 1 at 𝑥 = 0

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    Derivative of 𝑓(𝑥) = 𝑎^𝑥

    • 𝑓’(𝑥) = 𝑙𝑖𝑚_𝑑𝑥→0[(𝑎^{𝑥+𝑑𝑥}- 𝑎^𝑥) / 𝑑𝑥]
    • 𝑓’(𝑥) = 𝑙𝑖𝑚_𝑑𝑥→0[(𝑎^𝑥𝑎^𝑑𝑥- 𝑎^𝑥) / 𝑑𝑥]
    • 𝑓’(𝑥) = 𝑙𝑖𝑚_𝑑𝑥→0[𝑎^𝑥(𝑎^𝑑𝑥- 1) / 𝑑𝑥]
    • 𝑓’(𝑥) = 𝑙𝑖𝑚_𝑑𝑥→0𝑎^𝑥[(𝑎^𝑑𝑥- 1) / 𝑑𝑥]
    • 𝑓’(𝑥) = 𝑎^𝑥· 𝑙𝑖𝑚_𝑑𝑥→0[(𝑎^𝑑𝑥- 1) / 𝑑𝑥]

    Now plug in a value for 𝑎

    • 𝑓(𝑥) = 2^𝑥 then:
      • 𝑓’(𝑥) = 2^𝑥· 𝑙𝑖𝑚_𝑑𝑥→0[(2^𝑑𝑥- 1) / 𝑑𝑥]
      • 𝑓’(𝑥) = 2^𝑥· 0.69314718056…
    • 𝑓(𝑥) = 𝑒^𝑥 then:
      • 𝑓’(𝑥) = 𝑒^𝑥· 𝑙𝑖𝑚_𝑑𝑥→0[(𝑒^𝑑𝑥- 1) / 𝑑𝑥]
      • 𝑓’(𝑥) = 𝑒^𝑥· 1
      • 𝑓’(𝑥) = 𝑒

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