References: Why e is special

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Calculus, Chapter 6: "What's so special about Euler's number e?"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/eulers-number
  Series index: https://www.3blue1brown.com/?topic=calculus
  License: copyright Grant Sanderson; videos published on his site and YouTube
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Watch this next

• What’s so special about Euler’s number e? (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Watching the multiplier on a^x slide from below 1 (base 2) to above 1 (base 3) and pass through exactly 1 makes the definition of e feel inevitable rather than arbitrary, and the “rate proportional to value” framing connects it to growth and decay vividly. About fourteen minutes.

Going deeper

• Essence of Calculus (full series) by 3Blue1Brown. The series this track follows. The previous chapter gave the chain rule used here on e^(kx); the next (Implicit differentiation, what’s going on here?) differentiates relationships that are not solved for one variable.

• Khan Academy: Calculus for a slower, exercise-driven treatment of e, the natural logarithm, and the derivative of exponential functions, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track and the wider curriculum.

• The chain rule (previous-but-one lesson). The chain rule combined with the self-derivative property gives d/dx(e^(kx)) = k·e^(kx), the solution to “rate proportional to value.” Without the chain rule, e would only differentiate cleanly in the bare e^x case.

• Taylor series (final lesson). The Taylor series of e^x is 1 + x + x^2/2! + x^3/3! + ..., and it can be derived directly from the self-derivative property: a function equal to its own derivative forces exactly that pattern of coefficients. The capstone lesson develops this, with e^x as the cleanest example.

• Softmax and activations (Track 11, neural networks). The exponential defined here is the engine of softmax and the sigmoid, which appear at the output and inside the layers of the networks Track 11 builds. This lesson is the calculus prerequisite for understanding why those functions behave and differentiate the way they do.