Answer:
\(\displaystyle \frac{dy}{dx} = -4 \sin(x) \cos(x)\)
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: \(\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)\)
Derivative Property [Addition/Subtraction]: \(\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]\)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]: \(\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)\)
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
\(\displaystyle y = \cos(x) \sin(x)\)
Step 2: Differentiate
- Derivative Rule [Product Rule]: \(\displaystyle y' = \frac{d}{dx}[\cos(x)] \sin(x) + \cos(x) \frac{d}{dx}[\sin(x)]\)
- Trigonometric Differentiation: \(\displaystyle y' = -\sin^2(x) + \cos^2(x)\)
- Derivative Property [Addition/Subtraction]: \(\displaystyle y' = \frac{d}{dx}[-\sin^2(x)] + \frac{d}{dx}[\cos^2(x)]\)
- Rewrite [Derivative Property - Multiplied Constant]: \(\displaystyle y' = -\frac{d}{dx}[\sin^2(x)] + \frac{d}{dx}[\cos^2(x)]\)
- Basic Power Rule [Derivative Rule - Chain Rule]: \(\displaystyle y' = -2 \sin(x) \frac{d}{dx}[\sin(x)] + 2 \cos(x) \frac{d}{dx}[\cos(x)]\)
- Trigonometric Differentiation: \(\displaystyle y' = -2 \sin(x) \cos(x) - 2 \cos(x) \sin(x)\)
- Simplify: \(\displaystyle y' = -4 \sin(x) \cos(x)\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
