Answer:
[tex]\displaystyle F'(t) = 2e^{8t} \bigg( \cos (2x) + 4 \sin (2x) \bigg)[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle F(t) = e^{8t} \sin 2t[/tex]
Step 2: Differentiate
- Derivative Rule [Product Rule]: [tex]\displaystyle F'(t) = (e^{8t})' \sin 2t + e^{8t}(\sin 2t)'[/tex]
- Exponential Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle F'(t) = e^{8t}(8t)' \sin 2t + e^{8t}(\sin 2t)'[/tex]
- Trigonometric Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle F'(t) = e^{8t}(8t)' \sin 2t + e^{8t} \cos 2t (2t)'[/tex]
- Basic Power Rule [Derivative Property - Multiplied Constant]: [tex]\displaystyle F'(t) = 8e^{8t} \sin 2t + 2e^{8t} \cos 2t[/tex]
- Factor: [tex]\displaystyle F'(t) = 2e^{8t} \bigg( \cos (2x) + 4 \sin (2x) \bigg)[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
