Lecture 6 · Barnett ch. 2 · about 40 min

The Derivative: Slopes, Rates and Marginal Cost

What a derivative really measures, how to compute one from its definition and with the basic rules, and why economists call it the marginal cost or marginal revenue.

By the end you can

Compute an average rate of change as the slope of a secant line
Explain the derivative as the limit of secant slopes, and compute it from the definition
Differentiate polynomials with the constant, power, constant-multiple and sum rules
Use marginal cost, revenue and profit to estimate the effect of one more unit

How fast is it changing?

A firm’s cost rises from 5,000 to 5,900 lira when output goes from 100 to 110 units. On average, each extra unit cost 90 lira. That is an average rate of change: the change in the output of a function divided by the change in its input.

Average rate of changeFormula sheet →

\frac{f(a + h) - f(a)}{h}

$a$: starting input
$h$: change in the input
$f(a+h) - f(a)$: change in the output

Use it when a question asks for an average rate over an interval, such as average cost per extra unit between two output levels.

Geometrically, this is the slope of the straight line through the two points $(a, f(a))$ and $(a+h, f(a+h))$ . That line is called a secant line.

But a manager deciding whether to produce one more unit does not care about the average over the last ten. They want the rate of change right now, at the current output. That is what a derivative measures.

From secant to tangent

Make $h$ smaller and the second point slides toward the first. The secant line swings, and its slope settles on a single number: the slope of the tangent line, the line that just touches the curve at that point.

Try it. Start with $f(x) = x^2$ at $a = 1.5$ , then shrink $h$ and read the table.

A few things to notice:

For $x^2$ at $a = 1.5$ , the table approaches 3, which is $2 \times 1.5$ . Move $P$ and check that the limit is always $2a$ .
Switch to the cost function. The tangent slope at output $x$ is $x + 1$ : the cost of the next unit.
On $x^3 - 3x$ , put $P$ at a peak or a valley. The tangent is flat there, so the derivative is zero. You will use this for optimisation later in the course.

Definition of the derivativeDefinitionFormula sheet →

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

$f'(x)$: derivative of f at x, the instantaneous rate of change
$h \to 0$: the step shrinks toward zero but never equals it

Use it when a question says from the definition or using the limit. Otherwise use the rules below, which are much faster.

The derivative has three equivalent readings, and exams use all three words:

the slope of the tangent line to the graph at $x$ ,
the instantaneous rate of change of $f$ at $x$ ,
in economics, the marginal value: marginal cost, marginal revenue, marginal profit.

Computing a derivative from the definition

The definition always follows the same four steps. Learn the routine and exam questions of this type become mechanical.

The basic rules

Nobody computes derivatives from the definition every time. Four rules cover every polynomial.

Constant ruleFormula sheet →

\frac{d}{dx}\,k = 0

$k$: any constant

Use it when a term has no x. A flat line has slope zero, so fixed costs vanish from marginal cost.

Power ruleMost usedFormula sheet →

\frac{d}{dx}\,x^n = n\,x^{n-1}

$n$: any real number, including negatives and fractions

Use it when you see x to a power. Rewrite roots and fractions first: the square root of x is x to the 1/2, and 1/x is x to the -1.

Constant multiple and sum rulesFormula sheet →

\frac{d}{dx}\big[k\,f(x)\big] = k\,f'(x) \qquad \frac{d}{dx}\big[f(x) \pm g(x)\big] = f'(x) \pm g'(x)

$k$: constant
$f, g$: differentiable functions

Use it when always. Differentiate a polynomial term by term and keep the coefficients.

Marginal analysis

In economics the derivative gets a name of its own. If $C(x)$ is the total cost of producing $x$ units, then $C'(x)$ is the marginal cost. It approximates the cost of producing one more unit, the $(x+1)$ -th.

Marginal cost, revenue and profitEconomicsFormula sheet →

C'(x) \approx C(x+1) - C(x) \qquad P(x) = R(x) - C(x) \;\Rightarrow\; P'(x) = R'(x) - C'(x)

$C(x)$: total cost
$R(x)$: total revenue
$P(x)$: total profit
$x$: units produced

Use it when a question asks for the approximate cost, revenue or profit of the next unit. Use the exact difference only if it asks for the exact change.

Why only approximately? The marginal cost is the slope of the tangent at $x$ . The true cost of the next unit is the slope of the secant from $x$ to $x + 1$ : an $h = 1$ secant. When the curve bends slowly, the two are very close.

When there is no derivative

A derivative is a limit, and limits can fail to exist. At three kinds of points $f'(a)$ does not exist:

a corner, such as $|x|$ at $x = 0$ : secants from the left have slope $-1$ , from the right $+1$ ;
a vertical tangent, such as $\sqrt[3]{x}$ at $x = 0$ : the secant slopes grow without bound;
a break: if $f$ is not continuous at $a$ , it cannot be differentiable there.

Exam practice

Summary and review

Average rate of change $=$ slope of a secant $= \frac{f(a+h) - f(a)}{h}$ .
Derivative $=$ limit of secant slopes $=$ slope of the tangent $=$ instantaneous rate of change.
Four-step process: $f(x+h)$ , subtract $f(x)$ , divide by $h$ , let $h \to 0$ .
Rules: constants vanish; $x^n \to nx^{n-1}$ ; constants multiply through; differentiate sums term by term.
Rewrite roots and fractions as powers before differentiating.
$C'(x)$ , $R'(x)$ , $P'(x)$ approximate the change from one more unit.
Differentiable implies continuous, not the reverse.