Answer:
[tex]\displaystyle \frac{dA}{dt} = 8 \pi(16 \pi + 1)[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Calculus
Derivatives
Derivative Notation
Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Implicit Differentiation
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle A = 2\pi rh[/tex]
[tex]\displaystyle r = 2[/tex]
[tex]\displaystyle h = 4[/tex]
[tex]\displaystyle \frac{dr}{dt} = 16\pi[/tex]
[tex]\displaystyle \frac{dh}{dt} = 2[/tex]
Step 2: Differentiate
- [Implicit Differentiation] Product Rule: [tex]\displaystyle \frac{dA}{dt} = 2\pi \bigg( \frac{d}{dt}[r]h + r\frac{d}{dt}[h] \bigg)[/tex]
- Simplify: [tex]\displaystyle \frac{dA}{dt} = 2\pi \bigg( \frac{dr}{dt}h + r\frac{dh}{dt} \bigg)[/tex]
Step 3: Solve
- Substitute in variables [Derivative]: [tex]\displaystyle \frac{dA}{dt} = 2\pi \bigg( 16\pi(4) + 2(2) \bigg)[/tex]
- Evaluate: [tex]\displaystyle \frac{dA}{dt} = 8 \pi(16 \pi + 1)[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Implicit Differentiation
Book: College Calculus 10e
