Curriculum Overview: The Derivative as a Function

This curriculum module transitions students from the concept of a derivative at a specific point to the derivative function, $f'(x)$, which describes the instantaneous rate of change for all values in a function's domain.

## Prerequisites

Before beginning this module, students should have mastered the following concepts:

• Algebraic Manipulation: Proficiency in expanding binomials $(x+h)^n$ and rationalizing numerators.
• Functional Notation: Understanding $f(x)$ and the composition $f(x+h)$.
• The Limit of a Function: Ability to evaluate indeterminate forms, specifically $\frac{0}{0}$.
• Continuity: Knowledge of the three-part definition of continuity at a point $a$.
• Slope of a Tangent Line: Conceptual understanding of the derivative at a point $f'(a)$ as the limit of secant slopes.

## Module Breakdown

## Learning Objectives per Module

By the end of this curriculum, students will be able to:

Module 1: The Derivative Function

• Define the derivative function $f'(x)$ using the formal limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
• Apply the limit definition to find derivatives of polynomial, radical, and rational functions.

Module 2: Graphical Relationships

• Determine intervals where $f'(x) > 0$ (increasing $f$) and $f'(x) < 0$ (decreasing $f$).
• Identify where $f'(x) = 0 based on horizontal tangents on the graph of f(x)$.

Module 3: Continuity & Differentiability

• Theorem: Prove that if $f$ is differentiable at $a$, then $f$ is continuous at $a$.
• Identify non-differentiable points using the following visual triggers:

Module 4: Higher-Order Derivatives

• Calculate the second derivative $f''(x)$ or $\frac{d^2y}{dx^2}$.
• Understand the physical interpretation of higher orders (e.g., Acceleration as the derivative of Velocity).

## Success Metrics

To demonstrate mastery of "The Derivative as a Function," students must be able to:

• Calculate the derivative of $f(x) = \sqrt{x} using the limit definition, resulting in f'(x) = \frac{1}{2\sqrt{x}}$.
• Sketch a plausible graph of $f' given a complex graph of f$ containing at least one local maximum and one local minimum.
• Identify points of non-differentiability on a piecewise function (e.g., $f(x) = |x|$ at $x=0$).
• Translate fluently between Leibniz notation $\frac{dy}{dx}$ and prime notation $y'$.

## Real-World Application

Derivatives are not merely abstract limits; they represent the "speed" at which systems change.

[!IMPORTANT] The Physics Link: If $s(t) represents the position of a car, s'(t) is its **velocity**, and s''(t)$ (the second derivative) is its acceleration.

Case Study: Marginal Analysis in Economics

In business, the derivative of a Cost Function $C(x)$ is known as the Marginal Cost. It represents the cost of producing "one more unit."

Common Pitfalls to Avoid

• The Continuity Trap: Assuming that because a function is continuous, it must be differentiable. (Example: $f(x) = |x|$ is continuous at $x=0$ but NOT differentiable there).
• Notation Error: Confusing the second derivative notation $\frac{d^2y}{dx^2}$ with $(\frac{dy}{dx})^2$. The former is a derivative; the latter is a square.