Respuesta :

Space

Answer:

[tex]\displaystyle y' = \frac{3cos(2x) -2(3x + 1)[sin(2x) + cos(2x)]}{e^{2x}}[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

Algebra I

  • Factoring
  • Exponential Rule [Dividing]:                                                                         [tex]\displaystyle \frac{b^m}{b^n} = b^{m - n}[/tex]
  • Exponential Rule [Powering]:                                                                       [tex]\displaystyle (b^m)^n = b^{m \cdot n}[/tex]

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Product Rule:                                                                                                         [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]

Quotient Rule:                                                                                                       [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]

Trig Derivative:                                                                                                       [tex]\displaystyle \frac{d}{dx}[cos(u)] = -u'sin(u)[/tex]

eˣ Derivative:                                                                                                         [tex]\displaystyle \frac{d}{dx}[e^u] = u'e^u[/tex]

Step-by-step explanation:

Step 1: Define

[tex]\displaystyle y = \frac{(3x + 1)cos(2x)}{e^{2x}}[/tex]

Step 2: Differentiate

  1. [Derivative] Quotient Rule:                                                                           [tex]\displaystyle y' = \frac{\frac{d}{dx}[(3x + 1)cos(2x)]e^{2x} - \frac{d}{dx}[e^{2x}](3x + 1)cos(2x)}{(e^{2x})^2}[/tex]
  2. [Derivative] [Fraction - Numerator] eˣ derivative:                                       [tex]\displaystyle y' = \frac{\frac{d}{dx}[(3x + 1)cos(2x)]e^{2x} - 2e^{2x}(3x + 1)cos(2x)}{(e^{2x})^2}[/tex]
  3. [Derivative] [Fraction - Denominator] Exponential Rule - Powering:         [tex]\displaystyle y' = \frac{\frac{d}{dx}[(3x + 1)cos(2x)]e^{2x} - 2e^{2x}(3x + 1)cos(2x)}{e^{4x}}[/tex]
  4. [Derivative] [Fraction - Numerator] Product Rule:                                       [tex]\displaystyle y' = \frac{[\frac{d}{dx}[3x + 1]cos(2x) + \frac{d}{dx}[cos(2x)](3x + 1)]e^{2x} - 2e^{2x}(3x + 1)cos(2x)}{e^{4x}}[/tex]
  5. [Derivative] [Fraction - Numerator] [Brackets] Basic Power Rule:             [tex]\displaystyle y' = \frac{[(1 \cdot 3x^{1 - 1})cos(2x) + \frac{d}{dx}[cos(2x)](3x + 1)]e^{2x} - 2e^{2x}(3x + 1)cos(2x)}{e^{4x}}[/tex]
  6. [Derivative] [Fraction - Numerator] [Brackets] (Parenthesis) Simplify:       [tex]\displaystyle y' = \frac{[3cos(2x) + \frac{d}{dx}[cos(2x)](3x + 1)]e^{2x} - 2e^{2x}(3x + 1)cos(2x)}{e^{4x}}[/tex]
  7. [Derivative] [Fraction - Numerator] [Brackets] Trig derivative:                   [tex]\displaystyle y' = \frac{[3cos(2x) -2sin(2x)(3x + 1)]e^{2x} - 2e^{2x}(3x + 1)cos(2x)}{e^{4x}}[/tex]
  8. [Derivative] [Fraction - Numerator] Factor:                                                   [tex]\displaystyle y' = \frac{e^{2x}[(3cos(2x) -2sin(2x)(3x + 1)) - 2(3x + 1)cos(2x)]}{e^{4x}}[/tex]
  9. [Derivative] [Fraction] Simplify [Exponential Rule - Dividing]:                     [tex]\displaystyle y' = \frac{3cos(2x) -2sin(2x)(3x + 1) - 2(3x + 1)cos(2x)}{e^{2x}}[/tex]
  10. [Derivative] [Fraction - Numerator] Factor:                                                   [tex]\displaystyle y' = \frac{3cos(2x) -2(3x + 1)[sin(2x) + cos(2x)]}{e^{2x}}[/tex]

Topic: AP Calculus AB/BC

Unit: Derivatives

Book: College Calculus 10e

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