Differentiation Rules

Basic Differentiation Rules

The process of differentiation allows us to find the rate of change of a function with respect to its variable. Several fundamental rules simplify the calculation of derivatives for common types of functions.

• Derivative of a Constant Function: The derivative of any constant function is zero. where is a constant.

• Power Rule: For any real number , the derivative of is:

• Constant Multiple Rule: If is a constant and is differentiable:

• Sum and Difference Rules: The derivative of a sum or difference is the sum or difference of the derivatives:

Examples of Basic Differentiation

• Example 1:

• Example 2:

• Example 3:

Derivative of Exponential Functions

Exponential Differentiation

Exponential functions have unique differentiation properties, especially the natural exponential function .

• Derivative of :

• Derivative of :

Example

• Example:

Equations of Tangent Lines

Finding Tangent Lines

The tangent line to a curve at a given point is found using the derivative to determine the slope at that point.

• General Formula: For a function at point , the tangent line is:

• Example: at Tangent line:

Horizontal Tangents

Finding Horizontal Tangents

A tangent line is horizontal where the derivative equals zero.

• Example: Set :

Product Rule

Product Rule for Differentiation

The product rule is used when differentiating the product of two functions.

• Product Rule:

• Example:

Quotient Rule

Quotient Rule for Differentiation

The quotient rule is used when differentiating the quotient of two functions.

• Quotient Rule:

• Example:

Higher Derivatives

Second and Higher Derivatives

The derivative of a derivative is called a higher derivative. The second derivative measures the rate of change of the rate of change.

• Notation:

• Example:

Summary Table: Differentiation Rules