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Section3.1Motivation 3

Subsection3.1.1Derivatives: The Why

We could begin this discussion by focusing again on the demand function from our previous Decision Reports. What can we learn from how this function changes from one point to the next, that is, how a change in the independent variable quantity (\(q\)) effects a change in the dependent variable price (\(p = D(q)\))?

Click here to launch interactive.

In this interactive's default state it shows that, if \(q=400\) thousand units of my product are being produced, and if I were to increase my production quantity, I would need to decrease my unit price in order to sell them all. Specifically,

\begin{gather} \text{For each additional thousand units I produce, I'd have to drop the unit price by \$0.37.}\tag{3.1.1} \end{gather}

We call this a "marginal" analysis, or a computation of the marginal demand function.

Definition3.1.1Marginal function

Let \(f(q)\) be a function whose independent variable is \(q\text{,}\) the quantity of a product or service produced for sale.

The marginal function \(Mf(q)\) measures the amount this function's value would change as a result of increasing the production \(q\) by a single unit.

This is typically defined for \(f\) being demand, revenue, cost, or profit. For example, if \(f = R\) is a revenue function, then \(MR(60) = \) "at a production of \(q=60\) thousand units, how much revenue would be earned by producing 'one more' unit?"

Referring back to the marginal demand computation in (3.1.1), where we found in our example that \(MD(400) \approx -0.37\text{,}\) we'll see next that this quantity can be expressed and communicated in four important ways:

  1. Numerically, as a rate of change: "\(-\$0.37\) per thousand units of increase."
  2. Graphically, as the slope of the demand graph at \(q=400\text{,}\) or more specifically the slope of a line drawn tangent to the grpah at this point, as in the purple line below:
  3. In writing, as in (3.1.1), and lastly,
  4. In a formula, by deriving an expression for \(MD(q)\) and then evaluating it at \(q=400\text{.}\) Just how we get that formula is the subject of Section 3.2.

Subsection3.1.2Derivatives: The What, and the Rule of Four

Numerical: Derivative as a Rate of Change

Graphical: Derivative as a Slope

Verbal: Derivative as a Measure of Marginal Change

Algebraic: Derivative as a Function. If we follow the graphical approach of measuring the slope of a function's graph at each point, we can record those slopes at each point and thereby make a new function, called the derivative function

\begin{align*} f'(x) \amp\amp \text{or sometimes denoted }\frac{df}{dx} \end{align*}

read as "the derivative of \(f\) with respect to \(x\text{.}\)"

Because the derivative \(f'\) measures the slope of the graph of \(f\text{,}\) there will always be a close relationship between the values of the derivative \(f'\) and the direction of the function \(f\text{,}\) as you can see in the following interactive from PhET Sims:

Click here to launch interactive.