Answer:
\(\displaystyle y' = -6 \cos (3x) \sin (3x)\)
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: \(\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)\)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Chain Rule]: \(\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)\)
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle y = \cos^2 (3x)\)
Step 2: Differentiate
- Basic Power Rule [Derivative Rule - Chain Rule]: \(\displaystyle y' = 2 \cos (3x) \Big( \cos (3x) \Big)'\)
- Trigonometric Differentiation [Derivative Rule - Chain Rule]: \(\displaystyle y' = -2 \cos (3x) \sin (3x) (3x)'\)
- Rewrite [Derivative Property - Multiplied Constant]: \(\displaystyle y' = -6 \cos (3x) \sin (3x) (x)'\)
- Basic Power Rule: \(\displaystyle y' = -6 \cos (3x) \sin (3x)\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
