Question

Derivative of 2e^-1.5x*sin2pix

Answer 1

`\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 (d)/(dx) [cf(x)] = c \cdot f'(x)`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

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

Explanation:

Step 1: Define

Identify

`\displaystyle y = 2e^(-1.5x) \sin (2 \pi x)`

Step 2: Differentiate

1. Derivative Rule [Product Rule]:
   `\displaystyle y' = \big( 2e^(-1.5x) \big)' \sin (2 \pi x) + 2e^(-1.5x) \big( \sin (2 \pi x) \big)'`
2. 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)'`
3. Exponential Differentiation:
   `\displaystyle y' = -3e^(-1.5x) \sin (2 \pi x) + 2e^(-1.5x) \big( \sin (2 \pi x) \big)'`
4. 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)'`
5. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle y' = -3e^(-1.5x) \sin (2 \pi x) + 4 \pi e^(-1.5x) \cos (2 \pi x)(x)'`
6. Basic Power Rule:
   `\displaystyle y' = -3e^(-1.5x) \sin (2 \pi x) + 4 \pi e^(-1.5x) \cos (2 \pi x)`
7. Factor:
   `\displaystyle y' = e^(-1.5x) \bigg( -3 \sin (2 \pi x) + 4 \pi \cos (2 \pi x) \bigg)`
8. 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