Question

Derivative of Y=cos^2(3x)

Answer 1

`\displaystyle y' = -6 \cos (3x) \sin (3x)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

Explanation:

Step 1: Define

Identify

`\displaystyle y = \cos^2 (3x)`

Step 2: Differentiate

1. Basic Power Rule [Derivative Rule - Chain Rule]:
   `\displaystyle y' = 2 \cos (3x) \Big( \cos (3x) \Big)'`
2. Trigonometric Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle y' = -2 \cos (3x) \sin (3x) (3x)'`
3. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle y' = -6 \cos (3x) \sin (3x) (x)'`
4. Basic Power Rule:
   `\displaystyle y' = -6 \cos (3x) \sin (3x)`

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

Unit: Differentiation