Respuesta :

jcherry99

y = 1/(x - 2)   - 3

y' = d(x - 2)^-1 / dx   The three is a constant and will disappear.

y' = d(x - 2)^-1 -1 d(x - 2)/dx

y' = -1*(x - 2)^-2 * 1  

The minus comes from bringing down the original power down and reducing the power by one of the question. The 1 at the end comes from differentiating what is inside the brackets. It is called the chain rule

d(x - 2) / dx = d(x) / dx which is 1. The two, being a constant, disappears. The final answer is

y' = - (x - 2)^-2

Space

Answer:

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

Step 2: Differentiate

  1. Derivative Rule [Quotient Rule]:                                                                   [tex]\displaystyle y' = \frac{1'(x - 2) - 1(x - 2)'}{(x - 2)^2}[/tex]
  2. Basic Power Rule [Derivative Property - Addition/Subtraction]:               [tex]\displaystyle y' = \frac{-1(1)}{(x - 2)^2}[/tex]
  3. Simplify:                                                                                                         [tex]\displaystyle y' = \frac{-1}{(x - 2)^2}[/tex]

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

Unit: Differentiation