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Find the derivative of f(x) =( -7/x) at x = -3.

pls help !!

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Answer:

[tex]\displaystyle f'(-3) = \frac{7}{9}[/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

Calculus

Derivatives

Derivative Notation

Derivative of a constant is a 0

Basic Power Rule:

  • f(x) = cxⁿ
  • 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]

Step-by-step explanation:

Step 1: Define

[tex]\displaystyle f(x) = \frac{-7}{x}[/tex]

[tex]\displaystyle x = -3[/tex]

Step 2: Differentiate

  1. Quotient Rule:                                                                                                   [tex]\displaystyle f'(x) = \frac{\frac{d}{dx}[7]x - \frac{d}{dx}[x](-7)}{x^2}[/tex]
  2. Basic Power Rule:                                                                                             [tex]\displaystyle f'(x) = \frac{(0)x - x^{1 - 1}(-7)}{x^2}[/tex]
  3. Simplify:                                                                                                             [tex]\displaystyle f'(x) = \frac{(0)x - x^0(-7)}{x^2}[/tex]
  4. Simplify:                                                                                                             [tex]\displaystyle f'(x) = \frac{(0)x - 1(-7)}{x^2}[/tex]
  5. Multiply:                                                                                                             [tex]\displaystyle f'(x) = \frac{0 + 7}{x^2}[/tex]
  6. Add:                                                                                                                   [tex]\displaystyle f'(x) = \frac{7}{x^2}[/tex]

Step 3: Solve

  1. Substitute in x [Derivative]:                                                                              [tex]\displaystyle f'(-3) = \frac{7}{(-3)^2}[/tex]
  2. Evaluate exponents:                                                                                         [tex]\displaystyle f'(-3) = \frac{7}{9}[/tex]

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

Unit: Derivatives

Book: College Calculus 10e

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