Deriving the Exponential — e^x Is Its Own Slope
“eˣ is the only curve that is its own slope.”
The formula
d/dx eˣ = eˣHow to read it: eˣ differentiates to itself — its value is its own slope at every point
- eˣ
- — the exponential function with base e
- d/dx eˣ
- — its derivative (its slope)
- = eˣ
- — which is the same function again
The hook
The strangest, most beautiful function there is — eˣ differentiates into itself.
In plain words
The slope function of eˣ is eˣ again. At any point, the height there is exactly the slope there.
The intuition
The one curve that gets steeper the higher it is — and steeper by exactly its own height. It perfectly draws 'growth in proportion to current size', which is why nature's growth and decay all take this shape.
How it's built
This is why e was chosen as the 'base'. The derivative of any exponential aˣ is aˣ·ln a, and only when a=e does ln e=1 make the factor exactly 1, so it becomes itself.
Example
At x=0, e⁰=1 and the slope is 1. At x=1, e¹≈2.718 and the slope is 2.718. At every point, height = slope.
Common mistake
Not every exponential is its own derivative. The derivative of 2ˣ is 2ˣ·ln2 (ln2≈0.693), smaller than itself. The only base giving a factor of 1 is e.
Where it's used
eˣ is the solution to the differential equation y'=y ('rate of change proportional to current value') — the standard model for populations, interest, drug concentration, radioactivity.
Where it came from
When Euler set e as the natural base of the exponential, this 'self-derivative' property made e the star of calculus.
Prerequisites
Quick check
What is d/dx eˣ?
- x·eˣ⁻¹
- eˣ✓
- x·eˣ
- 1
Practice
What is d/dx eˣ evaluated at x=0?
Answer: 1
- Differentiating leaves eˣ unchanged
- Plug in x=0 → e⁰ = 1
Takeaway: The slope of eˣ is just its height at that point.
Differentiate y=e^(2x) and evaluate at x=0. (chain rule)
Answer: 2
- Outer eᵘ → eᵘ ; inner 2x → 2
- y'=2·e^(2x)
- Plug in x=0 → 2·1 = 2
Takeaway: The derivative of e^(kx) is k·e^(kx) — a factor of k appears.
Why isn't the derivative of 2ˣ equal to 2ˣ? Write on your notepad what differs from e.
Answer: undefined
- The derivative of aˣ is aˣ·ln a
- 2ˣ → 2ˣ·ln2, but ln2≈0.693≠1, so it isn't itself
- Only e has ln e=1, giving a factor of exactly 1
Takeaway: That factor of 1 is exactly what makes e special.