Calculus — Derivatives⏱ 5 min read

Exponential & Log Derivatives

d/dx(eˣ) = eˣ

What is the Exponential & Log Derivatives?

The number e ≈ 2.71828 is defined precisely so that its exponential function is its own derivative — d/dx(eˣ) = eˣ — which makes eˣ the single most important function in calculus. For a general base a, the derivative picks up an extra factor of ln a: d/dx(aˣ) = aˣ · ln a. The derivative of the natural logarithm is simply 1/x, for x > 0.

eˣ being its own derivative is why it shows up everywhere growth and decay are modeled — population growth, radioactive decay, compound interest, and more all reduce to equations built from eˣ.

What Each Variable Means

eˣ

Natural exponentiale ≈ 2.71828, defined precisely so that eˣ is its own derivative.

aˣ

General exponentialRequires a > 0 and a ≠ 1; its derivative picks up a factor of ln a.

ln x

Natural logarithmDefined only for x > 0; its derivative is 1/x.

When to Use It

Differentiating any exponential growth or decay expression
Differentiating logarithmic expressions, often combined with the product or quotient rule
Modeling rates of change in growth processes across physics, finance, and biology

Step-by-Step Examples

Example 1: A sum of exponentials

Problem: Differentiate y = 3eˣ + 5 · 2ˣ

Differentiate 3eˣ

Constants multiply straight through.

d/dx(3eˣ) = 3eˣ

Differentiate 5 · 2ˣ using the aˣ rule

Here a = 2, so the derivative picks up a factor of ln 2.

d/dx(5 · 2ˣ) = 5 · 2ˣ · ln 2

✓

Answer: dy/dx = 3eˣ + 5 · 2ˣ · ln 2

Example 2: A logarithm via the product rule

Problem: Differentiate y = x² · ln x

Identify f and g for the product rule

Split into the two factors.

f = x² → f' = 2x, g = ln x → g' = 1/x

Apply fg' + gf'

Combine according to the product rule.

x²·(1/x) + ln x·2x

Simplify

x²/x reduces to x.

x + 2x ln x

✓

Answer: dy/dx = x(1 + 2 ln x)

Interactive Calculator

x valuebase a (for aˣ)

Result will appear here

Common Mistakes

Mistake: Treating d/dx(aˣ) the same as d/dx(eˣ), forgetting the ln a factor.
Fix: Only base e gives back exactly itself. Any other base a picks up an extra factor of ln a: d/dx(aˣ) = aˣ · ln a.
Mistake: Using the power rule on eˣ, treating x as the base instead of the exponent.
Fix: eˣ has the variable in the exponent, not the base — the power rule (which needs a constant exponent) doesn't apply here at all.

Practice Questions

Differentiate y = 4ˣ.
Differentiate y = ln(x) + eˣ.

Frequently Asked Questions

Why is eˣ its own derivative?

That's essentially the defining property of e — it's the unique base for which the exponential function's growth rate exactly equals its own value at every point.

What's the derivative of ln(aˣ) for a general base a?

Using the log property ln(aˣ) = x ln a, this simplifies to a straight line, so its derivative is simply ln a — a constant.