EZ Math

Calculus · Derivatives

How to Use the Quotient Rule

Problem

f(x)=3x+1x−2f(x) = \frac{3x + 1}{x - 2}

Answer

f′(x)=−7(x−2)2f^{\prime}(x) = \frac{-7}{(x - 2)^2}

The quotient rule is the product rule with a minus sign and a squared denominator. Keep the subtraction order fixed to protect the sign.

Step-by-step solution

  1. Label the top as u and the bottom as v.

    u=3x+1,v=x−2u = 3x + 1,\quad v = x - 2

    Why: The quotient rule needs each part and its derivative.

  2. Differentiate each part.

    u′=3,v′=1u^{\prime} = 3,\quad v^{\prime} = 1

    Why: Both are linear, so each derivative is its coefficient.

  3. Write the quotient rule.

    f′(x)=u′v−uv′v2f^{\prime}(x) = \frac{u^{\prime} v - u v^{\prime}}{v^2}

    Why: The top is the derivative of the top times the bottom, minus the top times the derivative of the bottom.

  4. Substitute and simplify.

    f′(x)=3(x−2)−(3x+1)(x−2)2=3x−6−3x−1(x−2)2=−7(x−2)2f^{\prime}(x) = \frac{3(x - 2) - (3x + 1)}{(x - 2)^2} = \frac{3x - 6 - 3x - 1}{(x - 2)^2} = \frac{-7}{(x - 2)^2}

    Why: The x terms cancel and the constants combine.

Common mistakes

  • Swapping the order and using u v prime minus u prime v. That gives the wrong sign.
  • Forgetting to square the denominator.
  • Dropping the minus sign when subtracting 3x + 1.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. f(x)=x+1x−1f(x) = \frac{x + 1}{x - 1}

    Show answer

    f′(x)=−2(x−1)2f^{\prime}(x) = \frac{-2}{(x - 1)^2}

  2. f(x)=2xx+1f(x) = \frac{2x}{x + 1}

    Show answer

    f′(x)=2(x+1)2f^{\prime}(x) = \frac{2}{(x + 1)^2}

  3. f(x)=xx2+1f(x) = \frac{x}{x^2 + 1}

    Show answer

    f′(x)=1−x2(x2+1)2f^{\prime}(x) = \frac{1 - x^2}{(x^2 + 1)^2}

  4. f(x)=x2x−1f(x) = \frac{x^2}{x - 1}

    Show answer

    f′(x)=x2−2x(x−1)2f^{\prime}(x) = \frac{x^2 - 2x}{(x - 1)^2}

  5. f(x)=x+3xf(x) = \frac{x + 3}{x}

    Show answer

    f′(x)=−3x2f^{\prime}(x) = \frac{-3}{x^2}