Respuesta :

Space

Answer:

[tex]\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = -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  

Equality Properties

  • Multiplication Property of Equality
  • Division Property of Equality
  • Addition Property of Equality
  • Subtraction Property of Equality

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

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]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle V = \frac{1}{3} \pi x^2y[/tex]

[tex]\displaystyle V = 324 \pi[/tex]

[tex]\displaystyle x = 6[/tex]

[tex]\displaystyle y = 27[/tex]

Step 2: Differentiate

  1. Substitute in volume [Volume Formula]:                                                     [tex]\displaystyle 324 \pi = \frac{1}{3} \pi x^2y[/tex]
  2. [Equality Properties] Rewrite:                                                                        [tex]\displaystyle y = \frac{972}{x^2}[/tex]
  3. Quotient Rule:                                                                                               [tex]\displaystyle \frac{dy}{dx} = \frac{(972)'x^2 - (x^2)'972}{(x^2)^2}[/tex]
  4. Basic Power Rule:                                                                                         [tex]\displaystyle \frac{dy}{dx} = \frac{0x^2 - (2x)972}{(x^2)^2}[/tex]
  5. Simplify:                                                                                                         [tex]\displaystyle \frac{dy}{dx} = \frac{-1944x}{x^4}[/tex]
  6. Simplify:                                                                                                         [tex]\displaystyle \frac{dy}{dx} = \frac{-1944}{x^3}[/tex]

Step 3: Evaluate

  1. Substitute in variables [Derivative]:                                                             [tex]\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = \frac{-1944}{6^3}[/tex]
  2. Simplify:                                                                                                         [tex]\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = -9[/tex]

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

Unit: Differentiation

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