Chapter 3: Differentiation

3.1 Derivative of a Function

The derivative of a function measures how the function value changes as its input changes. It is a foundational concept in calculus, representing the instantaneous rate of change or the slope of the tangent line to the graph at a point.

• Definition: The derivative of the function f(x) with respect to the variable x is defined as:

• Geometric Interpretation: The derivative at a point gives the slope of the tangent line to the curve at that point.

• Notations: Common notations for derivatives include , , , and .

Example Problems (First Principles):

• at

Rules of Differentiation

Several rules simplify the process of finding derivatives for various types of functions.

• Rule 1: Derivative of a Constant Function

• Rule 2: Power Rule for Positive Integers

• Rule 3: Constant Multiple Rule

• Rule 4: Sum Rule

• Rule 5: Product Rule

• Rule 6: Quotient Rule

• Rule 7: Chain Rule If , then

Examples:

• at

Higher Order Derivatives

Derivatives of a function can be taken multiple times to obtain higher order derivatives, which provide information about the curvature and concavity of the function.

• First Derivative:

• Second Derivative:

• Third Derivative:

• n-th Derivative:

Examples:

3.2 Differentiation of Exponential and Logarithmic Functions

Exponential and logarithmic functions have unique differentiation rules due to their special properties.

• , where

• ,

• , where

Examples:

3.3 Derivatives of Trigonometric Functions

Trigonometric functions have well-defined derivatives that are essential in calculus and its applications.

Chain Rule for Trigonometric Functions:

Examples:

Proof Example:

3.4 Implicit Differentiation

Implicit differentiation is used when it is difficult or impossible to solve for one variable explicitly in terms of the other.

• Explicit Function:

• Implicit Function:

• Steps for Implicit Differentiation:

  1. Use the power rule for -terms.

  2. Use the chain rule for -terms (differentiate with respect to and multiply by ).

  3. Collect all terms on one side and solve for .

Examples:

3.5 Parametric Differentiation

Parametric differentiation is used when both and are given in terms of a third variable, called the parameter (often ).

• Parametric Equations: ,

• First Derivative:

• Second Derivative:

Examples:

• ,

• ,

• ,

3.6 Applications of Differentiation

Tangents and Normals

• Tangent Line: The tangent to the curve at has gradient and equation:

• Normal Line: The normal is perpendicular to the tangent and has gradient with equation:

Examples:

• Find the equation of the tangent and normal to at

• Find points on where the tangent is parallel to the -axis

• For , , find the tangent at

Extremum Problems and Critical Points

• Critical Point: A point where or is undefined.

• Stationary Point: Where (tangent is parallel to -axis).

• Non-differentiable Point: Where does not exist (tangent can be parallel to -axis or have a cusp).

• Critical Value: The -value at a critical point.

Examples:

Increasing and Decreasing Intervals

• Increasing: on interval

• Decreasing: on interval

• Constant: on interval

Example: For , find intervals of increase and decrease.

Relative Extremum and First Derivative Test

• Relative Maximum: If changes from positive to negative at , has a local maximum at .

• Relative Minimum: If changes from negative to positive at , has a local minimum at .

• First Derivative Test: Used to classify critical points as maxima, minima, or neither.

Examples:

• Find all relative extrema for

• Find maxima and minima for

Concavity and Point of Inflection

• Concave Upwards: (graph lies above tangents, gradient increases)

• Concave Downwards: (graph lies below tangents, gradient decreases)

• Point of Inflection: A point where the curve changes concavity (from up to down or vice versa)

Examples:

• For , determine intervals of concavity

• For , find points of inflection

• For , find first and second derivatives, critical points, and classify them

Second Derivative Test for Relative Extremum

• If , has a local minimum at

• If , has a local maximum at

• If , the test is inconclusive

Examples:

• Classify stationary points for

• Find maxima and minima for for