Derivatives of Trigonometric Functions

Introduction

In Business Calculus, understanding the derivatives of trigonometric functions is essential for solving problems involving rates of change, optimization, and modeling periodic phenomena. The six basic trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—each have specific rules for differentiation when the variable is in radians.

Basic Derivative Formulas

• Derivative of sine:

• Derivative of cosine:

• Derivative of tangent:

• Derivative of cotangent:

• Derivative of secant:

• Derivative of cosecant:

Chain Rule for Trigonometric Functions

When differentiating trigonometric functions of more complex arguments (e.g., or ), use the chain rule:

• General formula:

• Example:

Product and Quotient Rules with Trigonometric Functions

When trigonometric functions are multiplied or divided by other functions, apply the product rule or quotient rule:

• Product Rule:

• Quotient Rule:

• Example (Product):

• Example (Quotient):

Derivatives of Powers of Trigonometric Functions

• Derivative of : Use the chain rule:

• Derivative of :

• Derivative of :

Examples and Applications

• Example 1:

• Example 2:

• Example 3:

• Example 4:

• Example 5:

• Example 6:

• Example 7:

Table: Derivatives of the Six Trigonometric Functions

Summary

• Mastering the differentiation of trigonometric functions is crucial for solving calculus problems in business and economics.

• Always check if the function argument requires the chain rule.

• Use product and quotient rules when trigonometric functions are combined with other functions.

Additional info: Some examples and formulas have been expanded and clarified for academic completeness.