Differentiation Rules

Constant Function

The derivative of a constant function is always zero. This is because the graph of a constant function is a horizontal line, which has a slope of zero.

• Definition: If , then .

• Proof: Using the definition of the derivative:

Power Functions

Power functions are of the form , where is a positive integer. The derivative follows a simple rule known as the Power Rule.

• Power Rule:

• Examples:

• Proof (for general ): Using algebraic identities and the definition of the derivative:

Constant Multiple Rule

If a function is multiplied by a constant, the derivative is the constant times the derivative of the function.

• Rule:

• Example:

Sum and Difference Rules

The derivative of a sum or difference of functions is the sum or difference of their derivatives.

• Sum Rule:

• Difference Rule:

• Example:

Exponential Functions

The derivative of an exponential function depends on its base. The natural exponential function has a unique property: its derivative is itself.

• Rule:

• General Exponential: , where is the derivative at .

• Definition of :

• Example: If , then and .

Product Rule

The derivative of the product of two functions is given by the Product Rule.

• Rule:

• Example: If , then

• Higher Derivatives:

Quotient Rule

The derivative of a quotient of two functions is given by the Quotient Rule.

• Rule: ,

• Example:

• Compute ,

Table of Differentiation Formulas

The following table summarizes the main differentiation rules:

Applications of Differentiation Rules

Finding Horizontal Tangents

Horizontal tangents occur where the derivative of a function is zero.

• Example: For , set :

• Solutions:

• Corresponding points:

Motion: Position, Velocity, and Acceleration

In physics, derivatives are used to describe motion. The first derivative of position is velocity, and the second derivative is acceleration.

• Example:

• Velocity:

• Acceleration:

• Acceleration at : cm/s

Higher-Order Derivatives

Derivatives can be taken repeatedly to obtain higher-order derivatives, which are useful in various applications such as physics and engineering.

• Second Derivative:

• Third Derivative:

• nth Derivative:

• Example: For :

  • First:

  • Second:

  • Third:

  • Fourth:

Examples of Differentiation

• Power Rule with Negative and Fractional Exponents:

  • ,

  • ,

• Product Rule with Roots:

  • Rewrite:

  • Derivative:

• Quotient Rule:

Finding Tangent Lines

The equation of the tangent line to a curve at a given point can be found using the derivative.

• Example: at

• Derivative at :

• Tangent line:

• Example: at

• Derivative at : (horizontal tangent)

Summary Table: Differentiation Rules

Additional info: These notes cover Section 3.1 of a typical calculus textbook, focusing on differentiation rules for polynomials, exponential functions, and their applications. The content is expanded with academic context and examples for clarity and completeness.