Derivatives of Logarithmic Functions

Core Definition

For any real argument $x > 0$ : Natural Logarithm:

\frac{d}{dx}[\ln(x)] = \frac{1}{x}

General Logarithm (Base $a > 0, a \neq 1$ ):

\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}

When layered with a differentiable inner function

u = g(x) > 0

, the Chain Rule yields:

\frac{d}{dx}[\ln(u)] = \frac{1}{u}\frac{du}{dx} = \frac{u'}{u} \qquad \text{and} \qquad \frac{d}{dx}[\log_a(u)] = \frac{u'}{u \ln(a)}

Concept Explanation

1. Derivation via Implicit Differentiation

The derivative of $\ln(x)$ is established directly from the fact that $y = \ln(x)$ is the inverse of $x = e^y$ . Differentiating both sides implicitly with respect to $x$ :

\frac{d}{dx}[e^y] = \frac{d}{dx}[x] \implies e^y \frac{dy}{dx} = 1 \implies \frac{dy}{dx} = \frac{1}{e^y}

Because

e^y = x

, substituting back gives

\frac{dy}{dx} = \frac{1}{x}

2. General Logarithm via Change of Base

By the Change of Base formula (see Logarithmic and Exponential Properties), $\log_a(x) = \frac{\ln(x)}{\ln(a)}$ . Differentiating using the Constant Multiple Rule:

\frac{d}{dx}[\log_a(x)] = \frac{1}{\ln(a)} \cdot \frac{d}{dx}[\ln(x)] = \frac{1}{\ln(a)} \cdot \frac{1}{x} = \frac{1}{x \ln(a)}

3. Logarithmic Pre-Processing: The Time-Saving Technique

Before applying calculus derivative rules, always expand the argument completely using fundamental logarithm properties (see Logarithmic and Exponential Properties and Negative and Rational Exponents):

Product Property: $\ln(A \cdot B) = \ln(A) + \ln(B) \implies \frac{d}{dx}[\ln(AB)] = \frac{A'}{A} + \frac{B'}{B}$
Quotient Property: $\ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B) \implies \frac{d}{dx}\left[\ln\left(\frac{A}{B}\right)\right] = \frac{A'}{A} - \frac{B'}{B}$
Power Property: $\ln(A^k) = k \ln(A) \implies \frac{d}{dx}[\ln(A^k)] = k \frac{A'}{A}$

Differentiating the expanded sum of terms eliminates the need to execute cumbersome Quotient and Product rules inside $\frac{u'}{u}$ .

Engineering Applications

Aerospace Engineering (Tsiolkovsky Rocket Equation): The rocket velocity formula $v(m) = v_e \ln(m_0 / m) = v_e \ln(m_0) - v_e \ln(m)$ is differentiated to determine the vehicle's acceleration $\frac{dv}{dt} = -\frac{v_e}{m}\frac{dm}{dt}$ during fuel burn (see Rocket Velocity Optimization via Rocket Equation (Aerospace Engineering)).
Chemical Engineering (Acidity and pH Sensors): The pH of chemical solutions is tracked via $\text{pH} = -\log_{10}(H^+)$ . Calculating rate of pH change $\frac{d(\text{pH})}{dt} = -\frac{1}{H^+ \ln(10)}\frac{dH^+}{dt}$ provides feedback for automated buffering valves (see Reaction Rate Kinematics and pH Buffering (Chemical Engineering)).
Computer Science & Telecommunications (Entropy and Information): Shannon entropy and signal attenuation in communication channels depend on base-2 and natural logarithms.

Example Problems

Example 1: Differentiating with Pre-Processing

Differentiate $h(x) = \ln\left(\frac{x^2 - x}{x^2 + 2x}\right)$ for $x > 1$ .

Solution

Step 1: Simplify the argument algebraically Factor $x$ from numerator and denominator:

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

Step 2: Expand using the Quotient Property of logarithms

h(x) = \ln\left(\frac{x - 1}{x + 2}\right) = \ln(x - 1) - \ln(x + 2)

Step 3: Differentiate term-by-term using $\frac{d}{dx}[\ln(u)] = \frac{u'}{u}$

h'(x) = \frac{1}{x - 1} - \frac{1}{x + 2}

Step 4: Combine over a common denominator

h'(x) = \frac{(x + 2) - (x - 1)}{(x - 1)(x + 2)} = \frac{x + 2 - x + 1}{(x - 1)(x + 2)} = \frac{3}{(x - 1)(x + 2)}

Example 2: Product Rule with General Base Logarithm

Find the derivative of $g(x) = 3x^2 \log_3(x)$ .

Solution

Step 1: Identify the Product Rule structure Let $u = 3x^2$ and $v = \log_3(x)$ .

u' = 6x, \qquad v' = \frac{1}{x \ln(3)}

Step 2: Assemble the Product Rule

g'(x) = u'v + uv' = (6x)\log_3(x) + (3x^2)\left(\frac{1}{x \ln(3)}\right)

Step 3: Simplify algebraic factors

g'(x) = 6x \log_3(x) + \frac{3x}{\ln(3)}

Connections & References

Parent Concepts: Derivatives, Logarithmic and Exponential Properties
Prerequisites: Negative and Rational Exponents, Chain Rule
Inverse Companion: Derivatives of Exponential Functions
Advanced Continuation: Logarithmic Differentiation
Linked Engineering Applications: Rocket Velocity Optimization via Rocket Equation (Aerospace Engineering), Reaction Rate Kinematics and pH Buffering (Chemical Engineering)
Course Sessions: Calc-1 Session 08, Calc-1 Session 09