Question

Differentiate y = csc(x)cot(x).

Answer 1

`\displaystyle y' = - \csc (x) \big[ \cot^2 (x) + \csc^2 (x) \big]`

General Formulas and Concepts:

Algebra I

Terms/Coefficients

• Factoring

Functions

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Explanation:

Step 1: Define

Identify.

`\displaystyle y = \csc (x) \cot (x)`

Step 2: Differentiate

1. Derivative Rule [Product Rule]:
   `\displaystyle y' = (d)/(dx)[\csc (x)] \cot (x) + \csc (x) (d)/(dx)[\cot (x)]`
2. Trigonometric Differentiation:
   `\displaystyle y' = - \csc (x) \cot (x) \cot (x) + \csc (x) \big[ - \csc ^2 (x) \big]`
3. Simplify:
   `\displaystyle y' = - \csc (x) \cot^2 (x) - \csc^3 (x)`
4. Factor:
   `\displaystyle y' = - \csc (x) \big[ \cot^2 (x) + \csc^2 (x) \big]`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation