Mathematics · Year 12 · Further Calculus and Integration · Term 2

Derivatives of Logarithmic Functions

Students learn to differentiate logarithmic functions, including the natural logarithm.

• ACARA Content Descriptions
• AC9MFM07

About This Topic

Students differentiate logarithmic functions, with a focus on the natural logarithm ln(x), where the derivative is 1/x for x > 0. They extend this to composite functions using the chain rule, so the derivative of ln(g(x)) is [1/g(x)] * g'(x). Domain restrictions matter, since logarithms require positive arguments, and students analyze these limits in expressions.

This content aligns with AC9MFM07 in Further Calculus and Integration. It connects prior differentiation skills to advanced modelling, such as rates of change in exponential growth or decay scenarios common in biology and economics. Mastery here supports solving optimisation problems and prepares for integration by substitution.

Active learning benefits this topic because students often find the 1/x derivative counterintuitive compared to polynomial rules. When they use graphing software in pairs to zoom on ln(x) tangents or collaborate on chain rule puzzles, they see patterns emerge. Group derivations from limits build confidence, turning rote memorisation into deep understanding.

Key Questions

1. Explain the derivation of the derivative of ln(x).
2. Apply the chain rule to differentiate complex logarithmic expressions.
3. Analyze the domain restrictions that apply when differentiating logarithmic functions.

Learning Objectives

• Calculate the derivative of ln(x) using the limit definition.
• Apply the chain rule to differentiate composite logarithmic functions of the form ln(g(x)).
• Analyze domain restrictions for logarithmic functions and their derivatives.
• Synthesize the derivative rules for logarithmic functions to solve calculus problems.

Before You Start

The Chain Rule

Why: Students must be proficient with the chain rule to differentiate composite logarithmic functions.

Basic Differentiation Rules

Why: Students need a solid foundation in differentiating polynomial, exponential, and trigonometric functions before tackling logarithmic functions.

Properties of Logarithms

Why: Understanding logarithmic properties can simplify expressions before differentiation, making the process more manageable.

Key Vocabulary

Natural Logarithm	The logarithm to the base e, denoted as ln(x). It is the inverse function of the exponential function e^x.
Derivative of ln(x)	The rate of change of the natural logarithm function, which is 1/x for all x > 0.
Chain Rule	A calculus rule used to differentiate composite functions. If y = f(u) and u = g(x), then dy/dx = dy/du * du/dx.
Domain Restriction	The set of input values (x-values) for which a function is defined. For ln(x), the domain is x > 0.

Watch Out for These Misconceptions

Common Misconception

The derivative of ln(x) works for x ≤ 0.

What to Teach Instead

Logarithms are undefined for non-positive x, so derivatives apply only where x > 0. Active graphing tasks reveal vertical asymptotes at x=0, helping students visualise domain limits through peer comparison of plots.

Common Misconception

For ln(g(x)), the derivative is simply 1/g(x).

What to Teach Instead

The chain rule requires multiplying by g'(x). Relay activities break this into steps, so students practice each part collaboratively and catch omissions before full solutions.

Common Misconception

Derivative of log_b(x) is always 1/x, ignoring base.

What to Teach Instead

It is (1/(x ln b)). Jigsaw derivations let groups derive the change-of-base formula actively, reinforcing base effects through shared reconstruction.

Active Learning Ideas

Pair Graphing: Log Slopes

Pairs use graphing calculators to plot ln(x), ln(x^2), and ln(sin(x)) alongside their derivatives. They sketch tangents at x=1, x=2, and note slopes matching 1/x values. Discuss domain breaks and share findings with the class.

35 min·Pairs

Small Group Relay: Chain Rule Logs

Divide class into groups of four. Each member differentiates one step of a complex log expression, like d/dx [ln(3x^2 + 1)], then passes to the next for verification. Groups race to complete and present.

40 min·Small Groups

Jigsaw: Ln Derivation

Assign class sections the limit definition steps for ln(x)'s derivative. Groups assemble the proof on posters, then teach their part to others. Vote on clearest explanations.

45 min·Whole Class

Individual Exploration: Log Models

Students pick a real-world growth model, like bacterial populations N(t) = N0 e^{kt}, and find dN/dt using ln(N). They graph and interpret rates at different times.

25 min·Individual

Real-World Connections

• Biologists use logarithmic derivatives to model population growth rates, particularly in scenarios where growth is limited by resources. Understanding these rates helps predict species survival and manage ecosystems.
• Economists and financial analysts apply logarithmic functions and their derivatives to model compound interest and analyze the growth of investments over time. This informs financial planning and economic forecasting.
• Physicists use logarithmic relationships in describing phenomena like radioactive decay or the intensity of sound and light. The rate of change, found via derivatives, is crucial for understanding these physical processes.

Assessment Ideas

Quick Check

Present students with three functions: ln(x), ln(2x+1), and ln(x^2). Ask them to calculate the derivative for each and state the domain for which the derivative is valid. Review answers as a class, focusing on common errors with the chain rule and domain.

Exit Ticket

On a small card, ask students to write the derivative of f(x) = ln(sin(x)). Below their answer, they should write one sentence explaining the domain restriction for this derivative.

Discussion Prompt

Pose the question: 'Why is the derivative of ln(x) different from the derivative of x^n?' Facilitate a class discussion where students explain the derivation of the ln(x) derivative from first principles and compare it to polynomial differentiation rules.

Frequently Asked Questions

How do you derive the derivative of ln(x)?

Start from the limit definition: lim_{h→0} [ln(x+h) - ln(x)] / h = lim_{h→0} ln((x+h)/x) / h = (1/x) lim_{h→0} ln(1 + h/x) / h. Using ln(1+u)/u →1 as u→0, it simplifies to 1/x. Hands-on limit tables in pairs confirm this empirically before algebraic proof.

What are common mistakes with logarithmic derivatives?

Errors include ignoring domains, forgetting chain rule multipliers, or mishandling bases. Students plot functions and derivatives side-by-side to spot mismatches, like slopes not aligning at key points, building accuracy through visual feedback and discussion.

How to apply chain rule to log functions?

For ln(g(x)), differentiate as (1/g(x)) g'(x). Practice with u-substitution mentally: let u=g(x), du/dx=g'(x), d(ln u)/dx = (1/u) du/dx. Group relays sequence these steps, ensuring full application in compound expressions.

How can active learning help students master derivatives of logarithmic functions?

Active methods like paired graphing and group relays make abstract rules tangible. Students verify 1/x slopes visually, derive proofs collaboratively, and apply to models, reducing errors from rote learning. This builds intuition for chain rule and domains, with peer teaching solidifying recall for exams.