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r = \frac{a { }^(2) + 2a - 7 }{ √(a) }

differentiate this using the quotient rule




1 Answer

1 vote

Answer:


(dr)/(da) = \frac{3a^2 + 2a + 7}{2a^{(3)/(2) }}

General Formulas and Concepts:

Pre-Algebra

  • Distributive Property

Algebra I

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

Algebra II

  • Exponential Rule:
    x^(-m)= (1)/(x^m)
  • Exponential Rule:
    √(x) = x^{(1)/(2) }

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:
\displaystyle (d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))

Explanation:

Step 1: Define


r = (a^2+2a-7)/(√(a))

Step 2: Rewrite


r = \frac{a^2+2a-7}{a^{(1)/(2) }}

Step 3: Differentiate

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