Derivatives of Exponential and Logarithmic Functions

Exponential Functions

Natural Exponential

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

The function $e^x$ is its own derivative!

General Exponential

$\frac{d}{dx}[a^x] = a^x \cdot \ln(a)$

where $a > 0$, $a \neq 1$

With Chain Rule

$\frac{d}{dx}[e^u] = e^u \cdot u'$

$\frac{d}{dx}[a^u] = a^u \cdot \ln(a) \cdot u'$

Logarithmic Functions

Natural Logarithm

$\frac{d}{dx}[\ln(x)] = \frac{1}{x}, \quad x > 0$

General Logarithm

$\frac{d}{dx}[\log_a(x)] = \frac{1}{x \cdot \ln(a)}, \quad x > 0$

With Chain Rule

$\frac{d}{dx}[\ln(u)] = \frac{u'}{u}$

$\frac{d}{dx}[\log_a(u)] = \frac{u'}{u \cdot \ln(a)}$

Derivative Table

Function	Derivative
$e^x$	$e^x$
$a^x$	$a^x \ln(a)$
$\ln(x)$	$\frac{1}{x}$
$\log_a(x)$	$\frac{1}{x \ln(a)}$
$\ln	x
$e^u$	$e^u \cdot u'$
$\ln(u)$	$\frac{u'}{u}$

Important Property

$\frac{d}{dx}[\ln|x|] = \frac{1}{x} \quad \text{for all } x \neq 0$

This extends the domain to negative $x$ values.

Examples

Example 1: Basic exponential

$f(x) = 5e^x - 3e^{-x}$

$f'(x) = 5e^x - 3e^{-x} \cdot (-1) = 5e^x + 3e^{-x}$

Example 2: Exponential with chain rule

$f(x) = e^{x^2 + 3x}$

$f'(x) = e^{x^2 + 3x} \cdot (2x + 3)$

Example 3: General exponential

$f(x) = 2^x$

$f'(x) = 2^x \cdot \ln(2)$

Example 4: Natural logarithm

$f(x) = \ln(x^2 + 1)$

$f'(x) = \frac{2x}{x^2 + 1}$

Example 5: Product with logarithm

$f(x) = x \ln(x)$

$f'(x) = x \cdot \frac{1}{x} + \ln(x) \cdot 1 = 1 + \ln(x)$

Example 6: Quotient with exponential

$f(x) = \frac{e^x}{1 + e^x}$

$f'(x) = \frac{(1 + e^x) \cdot e^x - e^x \cdot e^x}{(1 + e^x)^2}$

$= \frac{e^x + e^{2x} - e^{2x}}{(1 + e^x)^2}$

$= \frac{e^x}{(1 + e^x)^2}$

Example 7: Logarithm of absolute value

$f(x) = \ln|\sin(x)|$

$f'(x) = \frac{\cos(x)}{\sin(x)} = \cot(x)$

Example 8: Change of base

$f(x) = \log_{10}(x) = \frac{\ln(x)}{\ln(10)}$

$f'(x) = \frac{1}{x \ln(10)}$