Question

Evaluate f′ (1) and f′′ (1): = x√x

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3√ 5

Answer 1

`\displaystyle f'(1) = (3)/(2)`

`\displaystyle f''(1) = (3)/(4)`

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 II

• 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

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

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

f(x) = x√x

f'(1) is x = 1 for 1st derivative

f''(1) is x = 1 for 2nd derivative

Step 2: Differentiate

1. [1st Derivative] Product Rule:
   `\displaystyle f'(x) = (d)/(dx)[x]√(x) + x(d)/(dx)[√(x)]`
2. [1st Derivative] Rewrite [Exponential Rule - Root Rewrite]:
   `\displaystyle f'(x) = (d)/(dx)[x]√(x) + x(d)/(dx)[x^{(1)/(2)}]`
3. [1st Derivative] Basic Power Rule:
   `\displaystyle f'(x) = (1 \cdot x^(1 - 1))√(x) + x((1)/(2)x^{(1)/(2)-1})`
4. [1st Derivative] Simply Exponents:
   `\displaystyle f'(x) = (1 \cdot x^0)√(x) + x((1)/(2)x^{(-1)/(2)})`
5. [1st Derivative] Simplify:
   `\displaystyle f'(x) = √(x) + x((1)/(2)x^{(-1)/(2)})`
6. [1st Derivative] Rewrite [Exponential Rule - Rewrite]:
   `\displaystyle f'(x) = √(x) + x(\frac{1}{2x^{(1)/(2)}})`
7. [1st Derivative] Rewrite [Exponential Rule - Root Rewrite]:
   `\displaystyle f'(x) = √(x) + x((1)/(2√(x)))`
8. [1st Derivative] Multiply:
   `\displaystyle f'(x) = √(x) + (x)/(2√(x))`
9. [2nd Derivative] Rewrite [Exponential Rule - Root Rewrite]:
   `\displaystyle f'(x) = x^{(1)/(2)} + \frac{x}{2x^{(1)/(2)}}`
10. [2nd Derivative] Basic Power Rule/Quotient Rule [Derivative Property]:
    `\displaystyle f''(x) = (1)/(2)x^{(1)/(2) - 1} + \frac{(d)/(dx)[x](2x^{(1)/(2)}) - x(d)/(dx)[2x^{(1)/(2)}]}{(2x^{(1)/(2)})^2}`
11. [2nd Derivative] Simplify/Evaluate Exponents:
    `\displaystyle f''(x) = (1)/(2)x^{(-1)/(2)} + \frac{(d)/(dx)[x](2x^{(1)/(2)}) - x(d)/(dx)[2x^{(1)/(2)}]}{4x}`
12. [2nd Derivative] Rewrite [Exponential Rule - Rewrite]:
    `\displaystyle f''(x) = \frac{1}{2x^{(1)/(2)}} + \frac{(d)/(dx)[x](2x^{(1)/(2)}) - x(d)/(dx)[2x^{(1)/(2)}]}{4x}`
13. [2nd Derivative] Basic Power Rule:
    `\displaystyle f''(x) = \frac{1}{2x^{(1)/(2)}} + \frac{(1 \cdot x^(1 - 1))(2x^{(1)/(2)}) - x((1)/(2) \cdot 2x^{(1)/(2) - 1})}{4x}`
14. [2nd Derivative] Simply Exponents:
    `\displaystyle f''(x) = \frac{1}{2x^{(1)/(2)}} + \frac{(1 \cdot x^0)(2x^{(1)/(2)}) - x((1)/(2) \cdot 2x^{(-1)/(2)})}{4x}`
15. [2nd Derivative] Simplify:
    `\displaystyle f''(x) = \frac{1}{2x^{(1)/(2)}} + \frac{2x^{(1)/(2)} - x((1)/(2) \cdot 2x^{(-1)/(2)})}{4x}`
16. [2nd Derivative] Multiply:
    `\displaystyle f''(x) = \frac{1}{2x^{(1)/(2)}} + \frac{2x^{(1)/(2)} - x(x^{(-1)/(2)})}{4x}`
17. [2nd Derivative] Rewrite [Exponential Rule - Rewrite]:
    `\displaystyle f''(x) = \frac{1}{2x^{(1)/(2)}} + \frac{2x^{(1)/(2)} - x(\frac{1}{x^{(1)/(2)}})}{4x}`
18. [2nd Derivative] Multiply:
    `\displaystyle f''(x) = \frac{1}{2x^{(1)/(2)}} + \frac{2x^{(1)/(2)} - \frac{x}{x^{(1)/(2)}}}{4x}`
19. [2nd Derivative] Simplify:
    `\displaystyle f''(x) = \frac{1}{2x^{(1)/(2)}} + \frac{x^{(1)/(2)}}{4x}`
20. [2nd Derivative] Rewrite [Exponential Rule - Root Rewrite]:
    `\displaystyle f''(x) = (1)/(2√(x)) + (√(x))/(4x)`

Step 3: Evaluate

1. [1st Derivative] Substitute in x:
   `\displaystyle f'(1) = √(1) + (1)/(2√(1))`
2. [1st Derivative] Evaluate Roots:
   `\displaystyle f'(1) = 1 + (1)/(2(1))`
3. [1st Derivative] Multiply:
   `\displaystyle f'(1) = 1 + (1)/(2)`
4. [1st Derivative] Add:
   `\displaystyle f'(1) = (3)/(2)`
5. [2nd Derivative] Substitute in x:
   `\displaystyle f''(1) = (1)/(2√(1)) + (√(1))/(4(1))`
6. [2nd Derivative] Evaluate Roots:
   `\displaystyle f''(1) = (1)/(2(1)) + (1)/(4(1))`
7. [2nd Derivative] Multiply:
   `\displaystyle f''(1) = (1)/(2) + (1)/(4)`
8. [2nd Derivative] Add:
   `\displaystyle f''(1) = (3)/(4)`