Curriculum Overview: Derivatives of Exponential and Logarithmic Functions

This curriculum covers the differentiation of transcendental functions, focusing on the unique properties of the natural base $e$ and the powerful technique of logarithmic differentiation. Students will learn to model rates of change in growth and decay scenarios.

## Prerequisites

Before beginning this module, students should have a firm grasp of the following concepts:

• Algebraic Foundations: Mastery of exponential laws ($b^x \cdot b^y = b^{x+y}$) and logarithmic properties (product, quotient, and power rules).
• The Chain Rule: Ability to differentiate composite functions, as most transcendental derivatives in practice involve $f(g(x))$.
• Implicit Differentiation: Necessary for understanding the derivation of logarithmic derivatives and for the technique of logarithmic differentiation.
• Basic Differentiation Rules: Proficiency with the Power, Product, and Quotient rules.

## Module Breakdown

## Learning Objectives per Module

Module 1: Natural Exponential and Logarithmic Functions

• Objective: Compute the derivative of $f(x) = e^{g(x)}$ and $f(x) = \ln(g(x))$.
• Insight: Understand why $e$ is the unique base where the slope of the tangent line equals the function value at every point.

Module 2: General Bases

• Objective: Apply the scaling factor $\ln b to find the derivative of b^x$ and $\log_b x$.
• Formula Box: $\frac{d}{dx} [b^x] = b^x \ln b \quad | \quad \frac{d}{dx} [\log_b x] = \frac{1}{x \ln b}$

Module 3: Logarithmic Differentiation

• Objective: Use $\ln to decompose complex functions like y = \frac{x^2 \sin x}{\sqrt{2x+1}}$ into sums and differences before differentiating.
• Process:
  1. Take $\ln$ of both sides.
  2. Expand using log properties.
  3. Differentiate implicitly.
  4. Solve for $y'$.

## Success Metrics

To demonstrate mastery of this curriculum, students must be able to:

• Differentiate any function of the form $a^{g(x)}$ without referencing a formula sheet.
• Identify when logarithmic differentiation is the most efficient strategy (e.g., variable bases like $x^x$ or complex rational products).
• Simplify logarithmic expressions using properties before applying the derivative operator to reduce computational error.
• Relate the derivative of a population model to its instantaneous growth rate at time $t$.

## Real-World Application

[!IMPORTANT] Why this matters: Exponential and logarithmic derivatives are the "language of growth."

• Biology: Modeling the spread of a virus where the rate of infection is proportional to the number of people already infected.
• Finance: Calculating the instantaneous rate of change in a continuously compounding investment account.
• Physics: Radioactive decay, where the rate of loss of mass is determined by the remaining amount of material ($N'(t) = -\lambda N(t)$).

[!TIP] Always check the domain of logarithmic functions. The derivative $\frac{d}{dx}\ln(g(x)) = \frac{g'(x)}{g(x)}$ is only valid where $g(x) > 0$.