Derivatives of Trigonometric Functions

Introduction to Trigonometric Derivatives

Trigonometric functions are fundamental in calculus, and their derivatives are essential for solving many problems involving rates of change and motion. Understanding the derivatives of these functions allows us to analyze their behavior and apply them in various contexts, such as physics and engineering.

• Trigonometric functions include sin x, cos x, tan x, csc x, sec x, and cot x.

• The derivative of a trigonometric function describes how the function changes with respect to its variable.

Basic Derivatives of Trigonometric Functions

The derivatives of the six primary trigonometric functions are as follows:

• Derivative of sin x:

• Derivative of cos x:

• Derivative of tan x:

• Derivative of csc x:

• Derivative of sec x:

• Derivative of cot x:

These formulas are used frequently in calculus to solve problems involving trigonometric functions.

Examples of Differentiation

Applying the rules above, we can differentiate more complex functions involving trigonometric terms.

• Example 1: Differentiate Solution:

  • Derivative of is

  • Derivative of (product rule):

  • Derivative of is

  Final answer:

• Example 2: Differentiate Solution: Use the quotient rule:

  • Numerator derivative:

  • Denominator derivative:

  • Quotient rule:

  Final answer:

• Example 3: Differentiate Solution: Use the product rule and derivatives of csc and cot:

  • Product rule:

  • Derivative of is

• Example 4: Differentiate Solution: Use the quotient rule:

  • Numerator derivative:

  • Denominator derivative:

  • Quotient rule:

Applications: Tangent Lines and Horizontal Tangents

Derivatives are used to find tangent lines and points where the tangent is horizontal (slope zero).

• Example: Find the equation of the tangent line to at Solution: The derivative is . Plug in to find the slope.

• Horizontal Tangents: For , set and solve for .

Motion Applications: Velocity and Speed

Trigonometric derivatives are used to analyze motion, such as the position, velocity, and speed of a particle.

• Example: If , then velocity .

• At : ft/s

• At : ft/s

• Speed is the absolute value of velocity:

Additional info: The notes also reference the product and quotient rules, which are essential for differentiating combinations of trigonometric functions. The product rule is , and the quotient rule is .