Respuesta :

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Answer:

[tex]\displaystyle f'(1) = \frac{3}{2}[/tex]

[tex]\displaystyle f''(1) = \frac{3}{4}[/tex]

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]:                                                                            [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

Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]

Derivative Rule [Product Rule]:                                                                             [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]

Derivative Rule [Quotient Rule]:                                                                           [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]

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

Step 3: Evaluate

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

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