Differentiation: Definition and Fundamental Properties Module Overview

Introduction

In this module, students build upon the concept of limits to formally define the derivative and explore its fundamental properties. The derivative represents the instantaneous rate of change of a function and is rooted in the limit definition. This module emphasizes interpreting derivatives graphically, numerically, analytically, and verbally while reinforcing proper mathematical notation and reasoning. Students will develop the ability to evaluate derivatives using foundational rules and apply their understanding to real-world contexts involving motion and change. Just as Unit 1 emphasized justifying conclusions about continuity and limits, this unit trains students to rigorously verify conditions before applying differentiation rules, fostering a deeper understanding of calculus as a system of logical reasoning built on definitions and theorems.

Essential Questions

How does the definition of the derivative emerge from the concept of a limit?
What does the derivative of a function tell us about the function’s behavior?
In what ways can we interpret and represent derivatives across different mathematical contexts?
How can we apply derivative rules to solve real-world problems involving change?
What is the difference between average and instantaneous rates of change?
Why is it important to verify the conditions of continuity and differentiability before applying a theorem?
How do algebraic, graphical, and numerical approaches to derivatives support one another?
How does the concept of differentiability relate to continuity?

Key Terms

The following key terms will help you understand the content in this module.

Rate of Change: How a quantity changes with respect to another (e.g., distance per time).

Instantaneous Rate of Change: The rate at which a function changes at a single point; this is the derivative.

Average Rate of Change: The change in the value of a function over an interval, calculated as the slope of the secant line. [f(b) - f(a)] / (b - a).

Derivative: A function that describes the instantaneous rate of change of another function; formally defined using a limit.

Difference Quotient: Expression used to calculate average rate of change or define a derivative. [f(x + h) - f(x)] / h

Limit Definition of the Derivative: The derivative as the limit of the difference quotient as h → 0.

f'(x): Leibniz notation for the derivative of function f with respect to x.

dy/dx: Derivative of y with respect to x using differential notation.

y': Alternate notation for the first derivative of y.

Tangent Line: A line that touches a curve at a single point and has the same slope as the function at that point.

Slope: A measure of the steepness of a line; in calculus, it often refers to the derivative.

Power Rule: d/dx[x^n] = n*x^(n-1)

Constant Rule: The derivative of a constant is 0.

Sum Rule: The derivative of a sum is the sum of the derivatives.

Difference Rule: The derivative of a difference is the difference of the derivatives.

Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative.

Continuity: A function is continuous if there are no breaks, jumps, or holes.

Differentiability: A function is differentiable at a point if the derivative exists there.

Corner: A sharp point on a graph where the derivative does not exist.

Cusp: A point where the slope approaches infinity from one or both sides.

Vertical Tangent: A tangent line with an undefined (infinite) slope.

Discontinuity: A point at which the function is not continuous.

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