Section 3.1: Introducing the Derivative

Tangent Lines

The concept of the tangent line is fundamental in calculus. The tangent line to a curve at a given point is the straight line that just "touches" the curve at that point and has the same instantaneous direction as the curve.

• Point of Tangency: The point (a, f(a)) where the tangent line touches the curve.

• Equation of a Line: To write the equation of a tangent line, we need a point and the slope at that point.

• Secant Line: A line passing through two points on the curve. As the two points get closer, the secant line approaches the tangent line.

• Limit Process: The slope of the tangent line is found as the limit of the slopes of secant lines as the two points merge.

Definition (Tangent Line): The tangent line to f at a is the line through (a, f(a)) with slope

or equivalently,

provided this limit exists.

Example 1

Find the equation of the tangent line to the graph of at .

• Find .

• Compute the slope using the limit definition:

• Equation:

Example 2

Find the equation of the tangent line to at .

• Find .

• Compute the slope:

• Equation:

The Derivative of a Function at a Point

The derivative of a function at a point measures the instantaneous rate of change of the function at that point.

• Definition: The derivative of at is

or

• The process of finding a derivative is called differentiation.

Example 3

For , find .

• Simplify numerator:

• So

Velocities and Rates of Change

Derivatives are used to describe rates of change in various contexts, such as physics.

• Position Function: gives the position of an object at time .

• Average Velocity:

• Instantaneous Velocity:

Example 4

A projectile is shot upward with initial velocity 120 ft/sec. Its position is .

• A. Average velocity between and :

• B. Average velocity between and :

• C. Average velocity between and :

• D. Instantaneous velocity at :

• , so ft/sec

Example 5

Given for , find the instantaneous rate of change at (midnight).

• At , °F/hour

Section 3.2: The Derivative as a Function

Derivative Function

The derivative function gives the slope of the tangent line to the graph of at any point where the derivative exists.

• Definition:

• Alternate Notations: , , ,

Example 1

Find the derivative of by the limit process.

• At , the slope is

Example 2

Find the derivative of by the limit process.

• At , the slope is

Graphing a Derivative

Given the graph of a function, the graph of its derivative shows the slope of the tangent line at each point.

• For , is a straight line through the origin with slope 2.

Derivatives and Continuity

• A function is differentiable at if exists.

• If is differentiable at , then is continuous at .

• However, continuity does not guarantee differentiability (e.g., corners, cusps, vertical tangents).

Example 4

Given , find .

• Check left and right limits for the derivative at .

Example 5

Is differentiable at ? No, because the graph has a vertical tangent at .

• Places where a function is not differentiable:

  • Corners and cusps

  • Discontinuities

  • Vertical tangent lines

Section 3.3: Rules of Differentiation

Basic Differentiation Rules

• Constant Rule: The derivative of a constant is 0.

• Power Rule: for any real number .

• Constant Multiple Rule:

• Sum and Difference Rules:

Examples

Derivative of the Natural Exponential Function

• For , (by product rule)

Higher-Order Derivatives

• The first derivative:

• The second derivative:

• The nth derivative:

Example

• For ,

Section 3.4: The Product and Quotient Rules

The Product Rule

If and are differentiable, then

Example

• For ,

The Quotient Rule

If and are differentiable and , then

• Mnemonic: "Low D High minus High D Low, all over Low squared"

Example

• For , apply the quotient rule.

Section 3.5: Derivatives of Trigonometric Functions

Basic Trigonometric Derivatives

Example

• (by chain rule)

Section 3.7: The Chain Rule

The Chain Rule

The chain rule is used to differentiate composite functions.

If , then

• Or, for

Example

• For ,

Section 3.8: Implicit Differentiation

Implicit Differentiation

Used when is not isolated on one side of the equation.

• Differentiate both sides with respect to .

• Collect all terms on one side.

• Solve for .

Example

• For , differentiate:

Logarithmic Differentiation

• Take the natural logarithm of both sides.

• Expand using log properties.

• Differentiate implicitly.

• Solve for .

Example

• For , , so , thus

Section 3.9: Derivatives of Logarithmic and Exponential Functions

Exponential and Logarithmic Derivatives

• for

Example

• For ,

• For ,