The Derivative of eˣ
Simple Explanation
The exponential function eˣ (where e≈2.71828 is Euler's number) is famously its own derivative: d/dx[eˣ] = eˣ.
Why Do We Need It?
This remarkable self-derivative property is exactly why e is chosen as the "natural" base for exponential functions in calculus — no other base has this simple property.
See It
An exponential curve with a straight tangent line touching it at the point (0,1)
Formula
The Derivative of eˣ
d/dx[eˣ] = eˣ
The exponential function with base e is famously its own derivative — the rate of change of eˣ, at any point, equals its own value there.
- e
- — Euler's number, approximately 2.71828
When to use it: Whenever differentiating an expression involving eˣ.
Worked Example
Differentiate an expression with eˣ
Find the derivative of f(x)=3eˣ, and evaluate f'(0).
Why Does This Work?
e is actually DEFINED (in one common approach) as the unique base for which this self-derivative property holds — the graph of eˣ, at every point, has a tangent line slope exactly equal to its own height there, exactly as shown at (0,1), where the height is 1 and the tangent slope is also 1.
Real-Life Example
Modeling continuously compounding interest
An investment account compounds interest continuously (not just yearly or monthly, but at every instant).
The eˣ function is exactly the natural model for continuous growth, and its self-derivative property makes its rate of growth calculations especially clean.
Practice
Find f'(0) for f(x)=eˣ+5.
MediumCommon mistake
Assuming the derivative of eˣ is something other than itself — eˣ is uniquely its own derivative, unlike any other exponential base.
Quick Review
- d/dx[eˣ] = eˣ.
- e is chosen specifically because it has this simple self-derivative property.
- At every point, eˣ's tangent slope equals its own height.