Derivatives of Exponential and Logarithmic Functions

Overview

The natural exponential function e^x has the remarkable property of being its own derivative. This single fact, combined with the chain rule and change-of-base formulas, generates the derivatives of every exponential and logarithmic function.

Intuition

e is defined precisely so that the exponential function e^x grows at a rate exactly equal to its own value at every point — that's what makes it 'natural.' Any other base a^x grows proportionally to a^x, but scaled by ln(a), a leftover factor from converting between bases.

Interactive Graph

Formal Definition

Definition

The core exponential and logarithmic derivatives:

\frac{d}{dx}e^x = e^x

\frac{d}{dx}a^x = a^x \ln(a) \quad (a>0)

\frac{d}{dx}\ln(x) = \frac{1}{x} \quad (x>0)

\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)} \quad (x>0,\ a>0,\ a\ne 1)

Derivation

Deriving d/dx ln(x) = 1/x using implicit differentiation, treating y = ln(x) as equivalent to e^y = x:

y = \ln(x) \iff e^y = x

Definition of natural log as inverse of e^x

\frac{d}{dx}\big[e^y\big] = \frac{d}{dx}[x]

Differentiate both sides with respect to x

e^y \cdot \frac{dy}{dx} = 1

Chain rule on the left side

\frac{dy}{dx} = \frac{1}{e^y} = \frac{1}{x}

Substitute back e^y = x

Applications

Continuously compounded interest A(t)=Pe^(rt) has A'(t)=rPe^(rt)=rA(t), meaning the growth rate is proportional to the current balance.

Worked Examples

Apply linearity and the standard derivatives (exp(x) = eˣ).
$f'(x) = 5e^x + \frac{3}{x}$

Answer: f'(x) = 5exp(x) + 3/x

Practice Problems

Difficulty 3/10

Differentiate f(x) = x²e^x (using the product rule).

Difficulty 4/10

Differentiate f(x) = ln(x²+1).

Common Mistakes

Common Mistake

Treating d/dx a^x as x·a^(x-1), applying the power rule to an exponential.

The power rule applies only when the variable is the base. For a^x (variable exponent, constant base), the derivative is a^x ln(a).

Common Mistake

Forgetting the ln(a) factor when differentiating log_a(x) or a^x for bases other than e.

Only base e avoids the extra ln(a) factor; d/dx e^x = e^x exactly, but d/dx a^x = a^x ln(a) for any other base.

Summary

d/dx e^x = e^x — e^x is the unique function (up to scalar multiples) that is its own derivative.
d/dx a^x = a^x ln(a) for general base a.
d/dx ln(x) = 1/x, derived via implicit differentiation from e^y = x.
d/dx log_a(x) = 1/(x ln(a)) via the change-of-base formula.

References

WebsiteWikipedia — Derivative of the exponential function