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[tex]r = \frac{a { }^{2} + 2a - 7 }{ \sqrt{a} } [/tex]
differentiate this using the quotient rule


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Sagot :

Spaceidn

Answer:

[tex]\frac{dr}{da} = \frac{3a^2 + 2a + 7}{2a^{\frac{3}{2} }}[/tex]

General Formulas and Concepts:

Pre-Algebra

  • Distributive Property

Algebra I

  • Expand by FOIL (First Outside Inside Last)
  • Terms/Coefficients/Degrees

Algebra II

  • Exponential Rule: [tex]x^{-m}= \frac{1}{x^m}[/tex]
  • Exponential Rule: [tex]\sqrt{x} = x^{\frac{1}{2} }[/tex]

Calculus

Derivatives

Derivative Notation

The derivative of a constant is equal to 0

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

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

[tex]r = \frac{a^2+2a-7}{\sqrt{a}}[/tex]

Step 2: Rewrite

[tex]r = \frac{a^2+2a-7}{a^{\frac{1}{2} }}[/tex]

Step 3: Differentiate

  1. Quotient Rule [Basic Power:                                                                          [tex]\frac{dr}{da} = \frac{a^{\frac{1}{2} }(2a^{2-1} + 2a^{1-1}) - \frac{1}{2}a^{\frac{1}{2} - 1 }(a^2 + 2a + 7)}{(a^{\frac{1}{2} })^2}[/tex]
  2. Simplify:                                                                                                            [tex]\frac{dr}{da} = \frac{a^{\frac{1}{2} }(2a + 2) - \frac{1}{2}a^{-\frac{1}{2}}(a^2 + 2a + 7)}{a}[/tex]
  3. Simplify:                                                                                                          [tex]\frac{dr}{da} = \frac{2}{2} \cdot \frac{a^{\frac{1}{2} }(2a + 2) - \frac{1}{2}a^{-\frac{1}{2}}(a^2 + 2a + 7)}{a}[/tex]
  4. Multiply:                                                                                                          [tex]\frac{dr}{da} = \frac{2a^{\frac{1}{2} }(2a + 2) - a^{-\frac{1}{2}}(a^2 + 2a + 7)}{2a}[/tex]
  5. Factor:                                                                                                             [tex]\frac{dr}{da} = \frac{a^{-\frac{1}{2}}[2a(2a + 2) - (a^2 + 2a + 7)]}{2a}[/tex]
  6. [Brackets] Distribute:                                                                                      [tex]\frac{dr}{da} = \frac{a^{-\frac{1}{2}}[4a^2 + 4a - a^2 - 2a - 7]}{2a}[/tex]
  7. [Brackets] Combine Like Terms:                                                                    [tex]\frac{dr}{da} = \frac{a^{-\frac{1}{2}}[3a^2 + 2a - 7]}{2a}[/tex]
  8. Simplify:                                                                                                            [tex]\frac{dr}{da} = \frac{3a^2 + 2a + 7}{2a^{\frac{3}{2} }}[/tex]
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