Euler's Number Math Example 2

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Example 2

hard
Show that the derivative of f(x)=exf(x) = e^x is itself, i.e., f′(x)=exf'(x) = e^x, using the limit definition of the derivative.

Solution

  1. 1
    Apply the limit definition: f′(x)=lim⁡h→0ex+h−exh=lim⁡h→0ex(eh−1)h=ex⋅lim⁡h→0eh−1hf'(x) = \lim_{h \to 0}\frac{e^{x+h} - e^x}{h} = \lim_{h \to 0}\frac{e^x(e^h - 1)}{h} = e^x \cdot \lim_{h \to 0}\frac{e^h - 1}{h}.
  2. 2
    Evaluate the remaining limit using the Taylor expansion eh=1+h+h22!+⋯e^h = 1 + h + \frac{h^2}{2!} + \cdots, so eh−1h=1+h2!+⋯→1\frac{e^h-1}{h} = 1 + \frac{h}{2!} + \cdots \to 1 as h→0h \to 0.
  3. 3
    Therefore f′(x)=ex⋅1=exf'(x) = e^x \cdot 1 = e^x. This self-referential property makes exe^x the unique exponential function equal to its own derivative.

Answer

ddxex=ex\frac{d}{dx}e^x = e^x
The fact that exe^x equals its own derivative is the defining characteristic of Euler's number. No other base axa^x satisfies ddxax=ax\frac{d}{dx}a^x = a^x without an extra multiplicative constant.

About Euler's Number

Euler's number e≈2.71828e \approx 2.71828 is the unique base for which the exponential function exe^x is its own derivative — the natural base for growth and decay.

Learn more about Euler's Number →

More Euler's Number Examples

Example 1 easy

Evaluate [formula] and state what this limit defines.

Example 3 easy

Using a calculator, compute [formula] to four decimal places. Then determine whether [formula].

Example 4 medium

A bank offers continuous compounding at an annual rate of [formula]. Using [formula], find how much

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