Lesson 12 The Product and Quotient Rules

12.1 Lecture Content

Video: The Product and Quotient Rules

12.2 Lecture Notes

12.2.1 Product Rule for Two Functions

If \(f(x)\) and \(g(x)\) are differentiable, then the derivative of their product is:

\[ \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \]

Mnemonic: “First times the derivative of the second, plus second times the derivative of the first.”

12.2.2 Product Rule for Three Functions

If \(f(x), g(x), h(x)\) are differentiable, then the derivative of their product is:

\[ \frac{d}{dx}[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) \]

Tip: Differentiate one function at a time, keeping the other two unchanged, and sum the results.

12.2.3 Quotient Rule

If \(f(x)\) and \(g(x)\) are differentiable and \(g(x) \ne 0\), then:

\[ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]

Mnemonic: “Low D high minus high D low, over low squared.”

12.3 Exercises

12.3.1 Applications of Differentiation

Video: Solutions

  1. Suppose a product’s revenue function is given by \[ R(q) = -3q^{2} + 800q. \] Find an expression for the marginal revenue function and simplify it.

  2. The R.C. Helliot Advertising Company finds that its profits on day \(t\) of an advertising campaign is given by \[ P(t) = -2.53t^{2} + 65t + 37,000, \] where \(P(t)\) is the profit in dollars for day \(t\).

    1. Find a simplified expression for the marginal profit function. Use the proper variable \(t\).
    2. What is the exact rate of change of profits when 44 days have passed in the campaign?
    3. What are the total profits on day 44 of the campaign?

  3. A simple item to be sold at a small specialty store has $200 in fixed costs and $8 per item in variable costs. The owners assume they can sell 8 items if they charge $28 each, and 5 items at a price of $55 each. (Assume that demand is a linear function.)

For how many units sold/produced is the marginal profit -124 dollars per item? (Use derivatives to find your answer.)

  1. In 1997, researchers at Texas A&M University estimated the operating costs of cotton gin plants of various sizes. A quadratic model of cost (in thousands of dollars) for the largest plants was found to be \[ C(q) = 0.021q^{2} + 21.2q + 387, \] where \(q\) is the annual quantity of bales (in thousands) produced by the plant. Revenue was estimated at $68 per bale of cotton.

Find the following:

  1. \(MC(q)\)
  2. The Marginal Revenue function
  3. The Marginal Profit function
  4. The Marginal Profit for \(q = 262\) thousand units
  5. What are the proper units for the answer to Part D?

12.3.2 The Product and Quotient Rules

Video: Solutions

  1. If \(f(t) = (t^2 + 2t + 5)(6t^2 + 5)\), find \(f'(t)\) and \(f'(4)\).

  2. Find the derivative of \(\cot(x) \cdot \frac{1}{x^3} (5x + 2)\).

  3. Find the derivative of \(f(x) = 16 \sqrt{x} \left(3x^4 - 3\right) \left(6x^8 + 7x\right)\).

  4. If \(f(x) = (3x^2 + 6x -5) \tan(x)\), find \(f'(x)\).

  5. If \(f(x) = \frac{\sqrt{x} - 4}{\sqrt{x} + 4}\), find \(f'(x)\) and \(f'(3)\).

  6. If \(f(x) = \frac{7x + 4}{7x + 5}\), find \(f'(x)\) and \(f'(5)\).

  7. If \(f(x) = \frac{7 - x^2}{2 + x^2}\), find \(f'(x)\)

  8. Find the first and second derivatives of \(f(x) = \frac{\sin(x)}{x^2}\).

12.4 Practice Problems

Practice the techniques discussed in class and in the online videos by solving the following examples.

  1. Differentiate:

    1. \(y = x^{2} \left( 4x^{3} + \frac{2}{x} - 1 \right)\).

    2. \(f(x) = x \ln x\).

    3. \(f(x) = (2x^{0.5} - x^{2})^{2}\).

  2. Differentiate:

    1. \(y = \sqrt{x} \left( 4x^{-1} - 2|x| - 10 \right)\).

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

    3. \(y = \left( \frac{x^{1.2}}{7} + \frac{2}{x^{2.1}} \right) \cos(x)\).

  3. Differentiate:

    1. \(f(x) = \frac{2x + 4}{3x - 1}\).

    2. \(y = \tan(x) = \frac{\sin(x)}{\cos(x)}\).

    3. \(f(x) = \frac{e^{2x}}{\ln(2x) + 2x}\).

  4. Differentiate:

    1. \(f(x) = \frac{(x-3)(x-2)(x-1)}{x+5}\).

    2. \(y = \frac{\sqrt{x} - 1}{\sqrt{x} + 1}\).

    3. \(y = \frac{(x+1)(x+2)}{(x-3)(x-2)(x-1)}\).

  5. (Applied) The monthly sales of Sunny Electronics’ new iSun walkman is given by \[ q(t) = 2000 t - 100 t^{2} \] units per month, \(t\) months after its introduction. The price Sunny charges is \[ p(t) = 100 - t^{2} \] dollars per iSun, \(t\) months after introduction. Find the rate of change of monthly sales, the rate of change of the price, and the rate of change of monthly revenue six months after the introduction of the iSun. Interpret your answers.

12.5 Self Assessment

Time yourself and try to solve the following questions within twenty minutes.

  1. Differentiate: \(f\left(x\right) = x^2\left(2x + 3\right)\left(7x + 2\right)\)
  2. Differentiate: \(y =\sin(x)\left(x^2 - 1\right)\)
  3. Differentiate: \(y =\dfrac{x}{\sqrt{x} + |x|}\)
  4. Differentiate: \(y =\dfrac{\frac{1}{x}-1}{\sin(x)}\)
  5. Thoroughbred Bus Company finds that its monthly costs for one particular year were given by \(C(t) = 100 + t^2\) dollars after \(t\) months. After $ t$ months, the company had \(P(t) = 1,000 + t^2\) passengers per month. How fast is its cost per passenger changing after 6 months?

12.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill D CON COM
Find the derivatives using the product rule.
Find the derivatives using the quotient rule.