Derivatives and Tangent Lines

Introduction

The concept of the derivative is fundamental in calculus, representing the instantaneous rate of change of a function. This study guide covers how to estimate slopes from graphs, calculate derivatives using the definition, and find equations of tangent lines to curves.

Estimating the Slope of a Curve at a Point

Understanding Slope and Tangency

• Slope at a Point: The slope of a curve at a specific point is the slope of the tangent line to the curve at that point.

• Tangent Line: A straight line that touches a curve at a single point without crossing it locally.

• Undefined Slope: If the tangent is vertical, the slope is undefined.

Example: If a curve has a sharp corner or a vertical tangent at a point, the slope at that point is undefined.

The Derivative: Definition and Calculation

Definition of the Derivative

• The derivative of a function f(x) at a point x = a is defined as:

• This limit, if it exists, gives the instantaneous rate of change of f at a.

Calculating the Derivative Using the Definition

• Given a function, substitute into the definition and simplify.

• Example: For , the derivative is:

• Thus, .

Evaluating the Derivative at Specific Points

• To find , substitute into the derivative:

• To find , substitute :

Equation of the Tangent Line

Finding the Equation of a Tangent Line

• The equation of the tangent line to the curve at the point is:

• Here, is the slope of the tangent at .

Examples

• For at :

Equation:

• For at :

Equation:

Summary Table: Derivative and Tangent Line Formulas

Key Points to Remember

• The slope of a curve at a point is the value of the derivative at that point.

• If the tangent is vertical or the function is not differentiable at a point, the slope is undefined.

• The equation of the tangent line uses the point-slope form with the derivative as the slope.

• Always use the definition of the derivative for rigorous calculation, especially when first learning the concept.