Why eˣ Differentiates to Itself
The exponential function f(x) = eˣ is the unique (up to a constant multiple) non-zero differentiable function that equals its own derivative. This remarkable property is not a coincidence — it follows directly from how the constant e is defined.
Step 1: The Difference Quotient for aˣ
For any base a > 0, apply the limit definition to f(x) = aˣ:
Factor out aˣ (which does not depend on h):
The remaining limit depends only on the base a. Call it L(a):
For a = 2, L(2) ≈ 0.693. For a = 3, L(3) ≈ 1.099. There is exactly one base for which L(a) = 1 — and that base is defined to be e.
Step 2: The Definition of e
The number e is defined as the unique positive real number satisfying:
Equivalently, e can be characterised as:
With L(e) = 1, the derivative of eˣ is simply:
Step 3: Confirmation via the Taylor Series
The Taylor series of eˣ centred at 0 can be derived by requiring the series Σ Cₙxⁿ to equal its own term-by-term derivative. This forces C(n+1)·(n+1) = Cₙ, with C₀ = e⁰ = 1. Solving the recurrence gives Cₙ = 1/n!, so:
Differentiating term by term (valid since the series converges uniformly on bounded intervals):
The series is unchanged by differentiation — a third confirmation that eˣ is its own derivative.
Derivative of aˣ for Other Bases
Using logarithmic differentiation, write aˣ = e^(x·ln a). By the chain rule:
When a = e, ln(e) = 1 and the formula reduces to d/dx eˣ = eˣ, consistent with everything above.
Worked Example
Differentiate f(x) = e^(3x²).
- Outer function: eᵘ, derivative eᵘ.
- Inner function: u = 3x², derivative 6x.
- Chain rule: f′(x) = e^(3x²) · 6x.
At x = 1: f′(1) = 6e³ ≈ 6 × 20.086 ≈ 120.5. Numerical verification with h = 10⁻⁶ confirms this value.
Frequently Asked Questions
Why is d/dx eˣ = eˣ?
Because e is precisely the base for which the exponential function equals its own derivative. Formally, e is defined so that lim_{h→0} (eʰ−1)/h = 1. This makes the difference quotient for eˣ evaluate to eˣ itself.
What is the Taylor series for eˣ?
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ··· = Σ_{n=0}^∞ xⁿ/n!. This series converges for all real (and complex) x. Differentiating term by term gives the same series, confirming d/dx eˣ = eˣ.
How is e defined as a limit?
e = lim_{n→∞} (1 + 1/n)ⁿ ≈ 2.71828... Equivalently, e = lim_{h→0} (1+h)^{1/h}. This limit arises naturally when computing continuous compound interest and in the definition of the natural logarithm.
What about d/dx aˣ for other bases?
For any positive base a ≠ 1, d/dx aˣ = aˣ·ln(a). When a = e, ln(e) = 1, so the formula reduces to d/dx eˣ = eˣ·1 = eˣ. The factor ln(a) measures how far a deviates from the 'natural' base e.
Adapted and rewritten from Silas Maths source notes; formulas reviewed for CalxSolver publication.