Question

Suppose the volume of the cone is 324pi Find dy/dx when x=6 and y=27

Answer 1

Answer:

$\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = -9$

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]:                                                                           $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle V = \frac{1}{3} \pi x^2y$

$\displaystyle V = 324 \pi$

$\displaystyle x = 6$

$\displaystyle y = 27$

Step 2: Differentiate

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

Step 3: Evaluate

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

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

Unit: Differentiation