Question

A function is given by f(x) =

`\sqrt[4]{x}`
Evaluate f'(16).​

Answer 1

`\displaystyle f'(16) = (1)/(32)`

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

Algebra I

• Exponential Rule [Rewrite]:
  `\displaystyle b^(-m) = (1)/(b^m)`
• Exponential Rule [Root Rewrite]:
  `\displaystyle \sqrt[n]{x} = x^{(1)/(n)}`

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Explanation:

Step 1: Define

`\displaystyle f(x) = \sqrt[4]{x}`

f'(16) is x = 16 for the derivative f'(x)

Step 2: Differentiate

1. [Function] Rewrite [Exponential Rule - Root Rewrite]:
   `\displaystyle f(x) = x^{(1)/(4)}`
2. Basic Power Rule:
   `\displaystyle f'(x) = (1)/(4)x^{(1)/(4) - 1}`
3. Simplify:
   `\displaystyle f'(x) = (1)/(4)x^{(-3)/(4)}`
4. Rewrite [Exponential Rule - Rewrite]:
   `\displaystyle f'(x) = \frac{1}{4x^{(3)/(4)}}`

Step 3: Solve

1. Substitute in x [Derivative]:
   `\displaystyle f'(16) = \frac{1}{4(16)^{(3)/(4)}}`
2. Evaluate exponents:
   `\displaystyle f'(16) = (1)/(4(8))`
3. Multiply:
   `\displaystyle f'(16) = (1)/(32)`

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

Unit: Derivatives

Book: College Calculus 10e