Answer:
[tex]\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = -9[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
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:
- 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
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
- Substitute in volume [Volume Formula]: [tex]\displaystyle 324 \pi = \frac{1}{3} \pi x^2y[/tex]
- [Equality Properties] Rewrite: [tex]\displaystyle y = \frac{972}{x^2}[/tex]
- Quotient Rule: [tex]\displaystyle \frac{dy}{dx} = \frac{(972)'x^2 - (x^2)'972}{(x^2)^2}[/tex]
- Basic Power Rule: [tex]\displaystyle \frac{dy}{dx} = \frac{0x^2 - (2x)972}{(x^2)^2}[/tex]
- Simplify: [tex]\displaystyle \frac{dy}{dx} = \frac{-1944x}{x^4}[/tex]
- Simplify: [tex]\displaystyle \frac{dy}{dx} = \frac{-1944}{x^3}[/tex]
Step 3: Evaluate
- Substitute in variables [Derivative]: [tex]\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = \frac{-1944}{6^3}[/tex]
- Simplify: [tex]\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = -9[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation