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Section 6.4: Derivatives of Logarithmic Functions

What This Section Is About and Why It Matters

This section answers a fundamental question: how do we differentiate logarithms? Since $\ln x$ is the inverse of $e^x$, and we already know $\frac{d}{dx}(e^x) = e^x$, we can find the derivative of $\ln x$ using implicit differentiation. The result, $\frac{d}{dx}(\ln x) = \frac{1}{x}$, is one of the cleanest formulas in calculus.

This matters because logarithms appear everywhere: in exponential growth and decay problems, in integration (where $\int \frac{1}{x}\,dx = \ln\vert x\vert + C$ fills a gap in the power rule), and in the powerful technique of logarithmic differentiation that simplifies products, quotients, and variable exponents.

Prerequisite Check

Before starting this section, can you...

Apply the chain rule?

Use implicit differentiation?

Work with logarithm properties?

Quick Reference: Key Formulas

Function	Derivative	Domain
$\ln x$	$\displaystyle\frac{1}{x}$	$x > 0$
$\ln\\|x\\|$	$\displaystyle\frac{1}{x}$	$x \neq 0$
$\ln(g(x))$	$\displaystyle\frac{g'(x)}{g(x)}$	$g(x) > 0$
$\log_a x$	$\displaystyle\frac{1}{x \ln a}$	$x > 0$, $a > 0$, $a \neq 1$

Integration Counterpart: $$\int \frac{1}{x}\,dx = \ln\vert x\vert + C$$

Dependency Map

graph TD
    subgraph Prerequisites
        CR["Chain Rule"]
        ID["Implicit<br/>Differentiation"]
        LP["Logarithm<br/>Properties"]
        DE["Derivative of e^x"]
    end

    subgraph "Section 6.4 Skills"
        DNL["<a href='derivative-natural-log.html'>Derivative of<br/>Natural Log</a>"]
        DGL["<a href='derivative-general-log.html'>Derivative of<br/>General Log</a>"]
        IYL["<a href='integral-yielding-ln.html'>Integrals<br/>Yielding ln</a>"]
        LD["<a href='logarithmic-differentiation.html'>Logarithmic<br/>Differentiation</a>"]
        EL["<a href='e-as-limit.html'>e as a Limit</a>"]
    end

    subgraph "Where This Leads"
        INT["Integration<br/>Techniques"]
        GROWTH["Exponential<br/>Growth Models"]
    end

    CR --> DNL
    ID --> DNL
    LP --> DNL
    DE --> DNL

    DNL --> DGL
    DNL --> IYL
    DNL --> LD
    DNL --> EL

    IYL --> INT
    LD --> INT
    EL --> GROWTH

    style DNL fill:#d1fae5,stroke:#10b981,stroke-width:2px
    style DGL fill:#dbeafe,stroke:#3b82f6,stroke-width:2px
    style IYL fill:#dbeafe,stroke:#3b82f6,stroke-width:2px
    style LD fill:#fef3c7,stroke:#f59e0b,stroke-width:2px
    style EL fill:#dbeafe,stroke:#3b82f6,stroke-width:2px

Skills in This Section

Core Skill

Skill	Description	Difficulty	Time
Derivative of Natural Log	The fundamental formula $\frac{d}{dx}(\ln x) = \frac{1}{x}$ and its proof	Intermediate	20 min

Building on the Core

Skill	Description	Difficulty	Time
Derivative of General Logarithms	Differentiating $\log_a x$ for any base $a$	Intermediate	15 min
Integrals Yielding ln	Recognizing when $\int f(x)\,dx = \ln\vert \cdot\vert + C$	Intermediate	25 min
e as a Limit	Understanding $e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n$	Intermediate	20 min

Advanced Technique

Skill	Description	Difficulty	Time
Logarithmic Differentiation	Using logs to differentiate products, quotients, and variable exponents	Advanced	25 min

Recommended Learning Path

If you're new to this material:

Start with Derivative of Natural Log: this is the foundation
Then Derivative of General Logarithms: extends the pattern
Next Integrals Yielding ln: the integration connection
Finally Logarithmic Differentiation: the powerful technique

If you're reviewing:

Jump directly to Logarithmic Differentiation for the most challenging technique
Use the Quick Reference table above for formula review

Common Mistakes in This Section

Mistake	Correction
Writing $\frac{d}{dx}(\ln x) = \frac{1}{\ln x}$	The derivative is $\frac{1}{x}$, not $\frac{1}{\ln x}$
Forgetting the chain rule: $\frac{d}{dx}[\ln(2x)] = \frac{1}{2x}$	Must include derivative of inside: $\frac{2}{2x} = \frac{1}{x}$
Writing $\int \frac{1}{x}\,dx = \ln x + C$	Need absolute value: $\ln\\|x\\| + C$ (valid for $x \neq 0$)
Confusing $(\ln x)^2$ with $\ln(x^2)$	$(\ln x)^2$: the log squared. $\ln(x^2) = 2\ln x$: log of a square

Section Mastery Checklist

By the end of Section 6.4, you should be able to:

[ ] State and prove $\frac{d}{dx}(\ln x) = \frac{1}{x}$ using implicit differentiation
[ ] Apply the chain rule: $\frac{d}{dx}[\ln(g(x))] = \frac{g'(x)}{g(x)}$
[ ] Differentiate logarithms with any base: $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$
[ ] Recognize integrals of the form $\int \frac{f'(x)}{f(x)}\,dx = \ln\vert f(x)\vert + C$
[ ] Use logarithmic differentiation on products, quotients, and variable exponents
[ ] Explain why $e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n$

| ← Section 6.3 | Skills Index | Section 6.5 → |