Medium

The Derivative of General Exponential Functions

Simple Explanation

For a base b other than e, the derivative of bˣ picks up an extra constant factor: d/dx[bˣ] = bˣ · ln(b).

Why Do We Need It?

Most real exponential models do not conveniently use base e directly (e.g. "doubling every hour" naturally uses base 2) — this formula handles any base correctly.

Formula

The Derivative of a General Exponential Function

d/dx[bˣ] = bˣ · ln(b)

For any base b, the derivative of bˣ is bˣ multiplied by an extra constant factor, ln(b) — this reduces to the simple eˣ case exactly when b=e, since ln(e)=1.

b: — the base of the exponential function
ln(b): — the natural logarithm of the base

When to use it: Whenever differentiating an exponential function with a base other than e.

Worked Example

Differentiate a general exponential function

Find the derivative of f(x)=2ˣ, and evaluate f'(0). (ln 2 ≈ 0.6931)

Why Does This Work?

Any base b can be rewritten as b=e^(ln b), so bˣ = e^(x·ln b) — differentiating this using the Chain Rule (with the eˣ derivative and the constant multiple ln b coming from the inner function's derivative) gives exactly bˣ·ln(b).

Real-Life Example

Finding the instantaneous growth rate of a doubling bacterial culture

A bacterial culture doubles in size every hour, modeled as P(t)=P₀·2ᵗ.

The general exponential derivative rule gives the instantaneous growth rate of the population at any specific time t.

Practice

For f(x)=bˣ with f'(x)=bˣ·ln(b), if b=e, what does ln(b) equal?

Medium

Common mistake

Forgetting the extra ln(b) factor for bases other than e — only b=e gives the simpler "derivative equals itself" result.

Quick Review

d/dx[bˣ] = bˣ · ln(b).
Reduces to the simple eˣ case exactly when b=e (since ln e=1).
Derived from rewriting bˣ as e^(x·ln b) and applying the Chain Rule.