Section 3.1

The Definition of the Derivative

Before shortcuts, we must understand the rigorous foundation: the derivative as the limit of the difference quotient.

Deep Dive: The Concept

Before learning the "shortcuts," we must understand the rigorous definition of a derivative.

Geometrically

The derivative represents the slope of the tangent line to a curve at a specific point. It is the instantaneous steepness.

Algebraically

It is the limit of the difference quotient (the slope of the secant line) as the distance between two points shrinks to zero.

If we have a function , the derivative is defined as:

This formula calculates the slope of the secant line between and , then pushes to 0 to find the tangent slope.

Find the derivative of using the limit definition.

Step 1: Set up the difference quotient

Step 2: Expand the numerator

Step 3: Simplify (cancel original terms)

Step 4: Factor out h and divide

Step 5: Evaluate the limit (h → 0)

Watch how the secant line (average rate of change) transforms into the tangent line (instantaneous rate of change) as approaches zero.

Function: f(x) = 3x^2 - x. Analyze at x = 1. Drag h to 0 to see the secant become the tangent.

Distance Limit (h): 1.00

Secant Slope (Average Rate)

8.0000

Tangent Slope (Instant Rate)

5.0000

Secant Line

Tangent Line