Answered

Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

Find the derivative of the function below in two ways. F(x) = x − 5x x x (a) by using the Quotient Rule F'(x) = (b) by simplifying first F'(x) = (c) Which method appears to be simpler for this problem? the Quotient Rule simplifying first

Sagot :

Space

Answer:

(a)  [tex]\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5[/tex]

(b)  [tex]\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5[/tex]

(c)  Simplifying first

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                          [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

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 [Basic Power Rule]:

  1. f(x) = cxⁿ
  2. 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]

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

Identify.

[tex]\displaystyle F(x) = \frac{x - 5x\sqrt{x}}{\sqrt{x}}[/tex]

Step 2: Differentiate Way 1

  1. Derivative Rule [Quotient Rule]:                                                                  [tex]\displaystyle F'(x) = \frac{\sqrt{x} \big( x - 5x\sqrt{x} \big) ' - \big( \sqrt{x} \big) ' \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}[/tex]
  2. Rewrite [Derivative Property - Addition/Subtraction]:                               [tex]\displaystyle F'(x) = \frac{\sqrt{x} \Big[ \big( x \big) '- \big( 5x\sqrt{x} \big) ' \Big] - \big( \sqrt{x} \big) ' \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}[/tex]
  3. Derivative Rule [Product Rule]:                                                                   [tex]\displaystyle F'(x) = \frac{\sqrt{x} \Big[ \big( x \big) '- \big( (5x)' \sqrt{x} + 5x(\sqrt{x})' \big) \Big] - \big( \sqrt{x} \big) ' \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}[/tex]
  4. Rewrite [Derivative Rule - Multiplied Constant]:                                        [tex]\displaystyle F'(x) = \frac{\sqrt{x} \Big[ \big( x \big) '- \big( 5(x)' \sqrt{x} + 5x(\sqrt{x})' \big) \Big] - \big( \sqrt{x} \big) ' \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}[/tex]
  5. Derivative Rule [Basic Power Rule]:                                                            [tex]\displaystyle F'(x) = \frac{\sqrt{x} \Big[ 1 - \big( 5\sqrt{x} + \frac{5x}{2\sqrt{x}} \big) \Big] - \frac{1}{2\sqrt{x}} \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}[/tex]
  6. Simplify:                                                                                                        [tex]\displaystyle F'(x) = \frac{\sqrt{x} \Big( 1 - 5\sqrt{x} - \frac{5x}{2\sqrt{x}} \Big) - \frac{x}{2\sqrt{x}} + \frac{5x\sqrt{x}}{2\sqrt{x}}}{x}[/tex]
  7. Simplify:                                                                                                        [tex]\displaystyle F'(x) = \frac{\sqrt{x} - 5x - \frac{5x}{2} - \frac{x}{2\sqrt{x}} + \frac{5x}{2}}{x}[/tex]
  8. Simplify:                                                                                                        [tex]\displaystyle F'(x) = \frac{\frac{\sqrt{x}}{2} - 5x}{x}[/tex]
  9. Simplify:                                                                                                        [tex]\displaystyle F'(x) = \frac{- \Big( 10\sqrt{x} - 1 \Big) }{2\sqrt{x}}[/tex]
  10. Rewrite:                                                                                                         [tex]\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5[/tex]

∴ we find the derivative of the given function but it is a tedious method of computation.

Step 3: Differentiate Way 2

  1. [Function] Rewrite:                                                                                       [tex]\displaystyle F(x) = \frac{x}{\sqrt{x}} - \frac{5x\sqrt{x}}{\sqrt{x}}[/tex]
  2. [Function] Simplify:                                                                                       [tex]\displaystyle F(x) = \sqrt{x} - 5x[/tex]
  3. [Derivative] Rewrite [Derivative Property - Addition/Subtraction]:           [tex]\displaystyle F'(x) = \big( \sqrt{x} \big) ' - \big( 5x \big) '[/tex]
  4. Rewrite [Derivative Property - Multiplied Constant]:                                 [tex]\displaystyle F'(x) = \big( \sqrt{x} \big) ' - 5 \big( x \big) '[/tex]
  5. Derivative Rule [Basic Power Rule]:                                                            [tex]\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5[/tex]

∴ we find the derivative of the given function and it is less complex and faster. We can conclude that simplifying first appears to be simpler for this problem.

---

Learn more about differentiation: https://brainly.com/question/17830594

Learn more about calculus: https://brainly.com/question/23558817

---

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

Unit: Differentiation

We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.