Most functions you work with grow by addition — a polynomial adds terms. Exponential functions grow by multiplication. That difference — multiplicative versus additive growth — produces something qualitatively different: growth that accelerates without bound, or decay that approaches zero but never reaches it. These functions appear constantly in physics, finance, and biology because they model any process where the rate of change is proportional to the current amount.

What Is an Exponential Function?

An exponential function has the form f(x) = aˣ where the base a is a positive constant and x is in the exponent. Unlike polynomials where the variable is the base, here the variable is the exponent — causing dramatically faster growth.

The Number e

The most important base for calculus is e ≈ 2.71828. It is the unique base where the exponential function equals its own derivative:

d/dx [eˣ] = eˣ
How e is Defined

e = lim(n→∞) (1 + 1/n)ⁿ. This limit arises naturally in continuously compounded interest — and turns out to be the most important constant in calculus.

Derivative Rules

d/dx [eˣ] = eˣ d/dx [e^f(x)] = e^f(x) · f'(x) d/dx [aˣ] = aˣ · ln(a)
📋 Worked Examples
1d/dx[e^(3x)] = 3e^(3x)
2d/dx[e^(x²)] = 2x·e^(x²)
3d/dx[2ˣ] = 2ˣ·ln(2)

Growth and Decay Models

Any quantity that grows or decays proportional to its size satisfies dy/dt = ky, with solution y = y₀·e^(kt). Population growth, radioactive decay, Newton's law of cooling, compound interest — all follow this pattern.

Frequently Asked Questions

Why is e special?▾
e is the unique number where the exponential function equals its own derivative. This makes calculus with base-e exponentials clean — any other base introduces an extra ln(a) factor in every derivative.
What is the integral of eˣ?▾
∫eˣ dx = eˣ + C. eˣ is its own antiderivative — a uniquely convenient property shared by no other function.

Where e Comes From — Compound Interest

Suppose you invest £1 at 100% annual interest. With annual compounding you end up with £2. With monthly compounding: (1 + 1/12)¹² ≈ £2.613. With daily compounding: (1 + 1/365)³⁶⁵ ≈ £2.7146. As the compounding frequency approaches infinity, the limit is exactly e ≈ 2.71828. This is why e appears naturally in any continuous growth process.

e = limn→∞ (1 + 1/n)n ≈ 2.71828...

The Self-Derivative Property

What makes e genuinely special — not just convenient, but mathematically fundamental — is that eˣ is its own derivative. This means if you measure the rate of change of eˣ at any point, you get eˣ back. No other function has this property. It makes eˣ the "natural" unit for any growth process where the rate of growth is proportional to the current amount.

d/dx [eˣ] = eˣ ← the only function equal to its own derivative

More Worked Derivatives

📋 Chain rule with exponentials
d/dx[e^(5x²)]= e^(5x²) · 10x = 10x·e^(5x²)
d/dx[e^(sin x)]= e^(sin x) · cos(x) = cos(x)·e^(sin x)
d/dx[3e^x + x²]= 3eˣ + 2x
d/dx[e^(x²+3x−1)]= e^(x²+3x−1) · (2x+3)

Integrating Exponentials

∫ eˣ dx = eˣ + C ∫ e^(ax) dx = e^(ax) / a + C (a ≠ 0) ∫ eˣ · f'(x) dx = eˣ · f(x) + C (by reverse product rule)
📋 Integration examples
∫ e^(3x) dx= e^(3x)/3 + C
∫ 2xe^(x²) dx= e^(x²) + C (u-sub: u = x²)
∫₀¹ eˣ dx= [eˣ]₀¹ = e − 1 ≈ 1.718

Real-World Exponential Models

Any quantity that grows or shrinks at a rate proportional to itself follows y = y₀eᵏᵗ. The constant k determines whether it grows (k > 0) or decays (k < 0):

📋 Common models
Radioactive decayN(t) = N₀e^(−λt). Half-life T = ln(2)/λ ≈ 0.693/λ
Population growthP(t) = P₀eʳᵗ where r is the net birth rate
Newton's coolingT(t) = Tₐ + (T₀−Tₐ)e^(−kt). Object cools toward ambient temperature Tₐ
Compound interestA = Pe^(rt) for continuous compounding at rate r
AM
Dr. Aisha Malik, PhD Mathematics
Senior Lecturer in Applied Mathematics · 12 years teaching calculus
Dr. Malik holds a PhD in Applied Mathematics from the University of Edinburgh and has taught calculus to over 4,000 students. She has reviewed this article for mathematical accuracy and pedagogical clarity.
Technically reviewed by: Prof. James Chen, Stanford Mathematics Department · April 2026