Respuesta :

Space

Answer:

[tex]\displaystyle u'(x) = \frac{-12}{(3x - 1)^2}[/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]

Derivative Property [Addition/Subtraction]:                                                        [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]  

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [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]

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 u(x) = \frac{4}{3x - 1}[/tex]

Step 2: Differentiate

  1. Derivative Rule [Quotient Rule]:                                                                   [tex]\displaystyle u'(x) = \frac{(4)'(3x - 1) - 4(3x - 1)'}{(3x - 1)^2}[/tex]
  2. Basic Power Rule [Derivative Properties]:                                                   [tex]\displaystyle u'(x) = \frac{-4(3)}{(3x - 1)^2}[/tex]
  3. Simplify:                                                                                                         [tex]\displaystyle u'(x) = \frac{-12}{(3x - 1)^2}[/tex]

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

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