Differentiation Rules

Basic Differentiation Rules

The process of differentiation allows us to find the rate of change of a function. 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:

Example:

Derivative of Exponential Functions

Exponential Rule

The exponential function is unique in that its derivative is itself. This property is fundamental in calculus and is often used in modeling growth and decay.

Example:

Equations of Tangent Lines

Finding Tangent Lines

The tangent line to a curve at a given point is a straight line that just touches the curve at that point and has the same slope as the curve there. The equation of the tangent line is found using the derivative evaluated at the point.

• General formula:

• Example: For at : Tangent line:

Horizontal Tangents

Finding Points with Horizontal Tangents

A tangent line is horizontal where the derivative of the function is zero. These points are often critical points in calculus.

• Example: For , set Corresponding points: and

Product Rule

Rule for Differentiating Products

The derivative of a product of two functions is not simply the product of their derivatives. The product rule states:

Example:

Quotient Rule

Rule for Differentiating Quotients

The derivative of a quotient of two functions is not simply the quotient of their derivatives. The quotient rule states:

Example:

Higher Derivatives

Second and Higher Order Derivatives

If is differentiable, its derivative is also a function and may itself be differentiated. The second derivative, , measures the rate of change of the rate of change, and is important in analyzing concavity and inflection points.

• Notation:

• Example: If :

Graphical Interpretation of Derivatives

Identifying Graphs of Functions and Their Derivatives

Graphs of a function, its first derivative, second derivative, and higher derivatives can be distinguished by their shapes and the number of inflection points or critical points. Polynomial functions of higher degree have more complex graphs, while their derivatives simplify the degree and shape.

• Odd degree polynomials: First derivative is even degree, second derivative is odd degree, etc.

• Even degree polynomials: First derivative is odd degree, second derivative is even degree, etc.