Answer:
\(\displaystyle y' = e^{-1.5x} \bigg( 4 \pi \cos (2 \pi x) - 3 \sin (2 \pi x) \bigg)\)
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 [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
\(\displaystyle y = 2e^{-1.5x} \sin (2 \pi x)\)
Step 2: Differentiate
- Derivative Rule [Product Rule]: \(\displaystyle y' = \big( 2e^{-1.5x} \big)' \sin (2 \pi x) + 2e^{-1.5x} \big( \sin (2 \pi x) \big)'\)
- Rewrite [Derivative Property - Multiplied Constant]: \(\displaystyle y' = 2 \big( e^{-1.5x} \big)' \sin (2 \pi x) + 2e^{-1.5x} \big( \sin (2 \pi x) \big)'\)
- Exponential Differentiation: \(\displaystyle y' = -3e^{-1.5x} \sin (2 \pi x) + 2e^{-1.5x} \big( \sin (2 \pi x) \big)'\)
- Trigonometric Differentiation [Derivative Rule - Chain Rule]: \(\displaystyle y' = -3e^{-1.5x} \sin (2 \pi x) + 2e^{-1.5x} \cos (2 \pi x)(2 \pi x)'\)
- Rewrite [Derivative Property - Multiplied Constant]: \(\displaystyle y' = -3e^{-1.5x} \sin (2 \pi x) + 4 \pi e^{-1.5x} \cos (2 \pi x)(x)'\)
- Basic Power Rule: \(\displaystyle y' = -3e^{-1.5x} \sin (2 \pi x) + 4 \pi e^{-1.5x} \cos (2 \pi x)\)
- Factor: \(\displaystyle y' = e^{-1.5x} \bigg( -3 \sin (2 \pi x) + 4 \pi \cos (2 \pi x) \bigg)\)
- Rewrite: \(\displaystyle y' = e^{-1.5x} \bigg( 4 \pi \cos (2 \pi x) - 3 \sin (2 \pi x) \bigg)\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
