Respuesta :

Space

Answer:

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

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

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 y = \frac{\cos (x - 1)}{x - 1}[/tex]

Step 2: Differentiate

  1. Derivative Rule [Quotient Rule]:                                                                   [tex]\displaystyle y' = \frac{\Big( \cos (x - 1) \Big)'(x - 1) - \cos (x - 1)(x - 1)'}{(x - 1)^2}[/tex]
  2. Trigonometric Differentiation [Derivative Rule - Chain Rule]:                   [tex]\displaystyle y' = \frac{- \sin (x - 1)(x - 1)'(x - 1) - \cos (x - 1)(x - 1)'}{(x - 1)^2}[/tex]
  3. Basic Power Rule [Derivative Properties]:                                                   [tex]\displaystyle y' = \frac{- \sin (x - 1)(x - 1) - \cos (x - 1)}{(x - 1)^2}[/tex]
  4. Simplify:                                                                                                         [tex]\displaystyle y' = \frac{- \cos (x - 1)}{(x - 1)^2} - \frac{\sin (x - 1)}{x - 1}[/tex]

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

Unit: Differentiation

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