Answer:
[tex]\displaystyle f'(16) = \frac{1}{32}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Algebra I
- Exponential Rule [Rewrite]: [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex]
- Exponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]
Calculus
Derivatives
Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Step-by-step explanation:
Step 1: Define
[tex]\displaystyle f(x) = \sqrt[4]{x}[/tex]
f'(16) is x = 16 for the derivative f'(x)
Step 2: Differentiate
- [Function] Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle f(x) = x^{\frac{1}{4}}[/tex]
- Basic Power Rule: [tex]\displaystyle f'(x) = \frac{1}{4}x^{\frac{1}{4} - 1}[/tex]
- Simplify: [tex]\displaystyle f'(x) = \frac{1}{4}x^{\frac{-3}{4}}[/tex]
- Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle f'(x) = \frac{1}{4x^{\frac{3}{4}}}[/tex]
Step 3: Solve
- Substitute in x [Derivative]: [tex]\displaystyle f'(16) = \frac{1}{4(16)^{\frac{3}{4}}}[/tex]
- Evaluate exponents: [tex]\displaystyle f'(16) = \frac{1}{4(8)}[/tex]
- Multiply: [tex]\displaystyle f'(16) = \frac{1}{32}[/tex]
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Derivatives
Book: College Calculus 10e
