Question

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

Answer 1

`\displaystyle (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 (d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))`

Explanation:

Step 1: Define

Identify

`\displaystyle V = (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 = (1)/(3) \pi x^2y`
2. [Equality Properties] Rewrite:
   `\displaystyle y = (972)/(x^2)`
3. Quotient Rule:
   `\displaystyle (dy)/(dx) = ((972)'x^2 - (x^2)'972)/((x^2)^2)`
4. Basic Power Rule:
   `\displaystyle (dy)/(dx) = (0x^2 - (2x)972)/((x^2)^2)`
5. Simplify:
   `\displaystyle (dy)/(dx) = (-1944x)/(x^4)`
6. Simplify:
   `\displaystyle (dy)/(dx) = (-1944)/(x^3)`

Step 3: Evaluate

1. Substitute in variables [Derivative]:
   `\displaystyle (dy)/(dx) \bigg| \limits_(x = 6) = (-1944)/(6^3)`
2. Simplify:
   `\displaystyle (dy)/(dx) \bigg| \limits_(x = 6) = -9`

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

Unit: Differentiation