Chapter 3: Derivatives

Section 3.3: Differentiation Rules

This section introduces fundamental rules for differentiating functions efficiently, without the need to compute limits for each case. Mastery of these rules is essential for solving a wide range of calculus problems.

Basic Derivative Rules

The following rules form the foundation for differentiating most functions encountered in Calculus I:

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

• Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.

• Power Rule: The derivative of is .

• Sum Rule: The derivative of a sum is the sum of the derivatives.

• Difference Rule: The derivative of a difference is the difference of the derivatives.

• Product Rule: The derivative of a product is given by:

• Quotient Rule: The derivative of a quotient is:

• Chain Rule: The derivative of a composite function is:

Examples: Differentiating Functions Using Rules

These examples illustrate the application of the basic derivative rules:

• Example 1: Apply the power rule:

• Example 2: Apply the constant rule:

• Example 3: Apply the constant multiple and power rules:

• Example 4: Differentiate term by term:

• Example 5: Rewrite as , then apply the power rule:

Product and Quotient Rule Applications

When functions are multiplied or divided, use the product or quotient rule:

• Example 6: Apply the quotient rule:

• Example 7: Apply the product rule:

Derivative of the Exponential Function

Exponential functions have unique differentiation properties:

• Example 8: Differentiate each term:

• Example 9: Apply the product rule:

Higher Order Derivatives

Higher order derivatives are obtained by repeatedly differentiating a function. The second derivative, , describes the rate of change of the rate of change, often used to analyze concavity and inflection points.

• Example 10: First derivative: Second derivative:

• Example 11: First derivative: Second derivative:

• Example 12: First derivative (product rule): Second derivative:

Derivatives of All Orders

Some functions allow for the computation of derivatives of all orders, which can be useful in Taylor series and advanced calculus applications.

• Example 13: First derivative: Second derivative: Third derivative: Fourth derivative:

Horizontal Tangent Lines

A curve has a horizontal tangent line where its derivative equals zero. This is often used to find local maxima, minima, or points of inflection.

• Example 14: Find and solve : Set $y' = 0$: Solutions: Thus, horizontal tangents occur at .