4 Derivative-Based Analysis

Rates of Change and Derivatives

A derivative measures how rapidly an output changes as its input changes. For a function $y=f(x)$, the derivative at an input is

This differs from the average rate of change over $[a,b]$:

The average rate describes change across an interval, whereas the derivative describes behavior at one input. Geometrically, $f'(a)$ is the slope of the tangent line to the graph at $x=a$.

Units provide an important interpretation check. A derivative has units of output divided by input. If $s(t)$ is position in meters and $t$ is time in seconds, then $s'(t)$ has units of meters per second and represents velocity, while $s''(t)$ has units of meters per second squared and represents acceleration.

A positive derivative indicates instantaneous growth, and a negative derivative indicates instantaneous decline. The magnitude describes how rapidly the change occurs, but its meaning depends on the quantities and units in the model.

Takeaway: A derivative is both a geometric slope and an instantaneous rate with units.

Differentiation Rules

Differentiation rules turn the limit definition into an efficient calculation method. For a constant $C$,

For any real number $n$, the power rule is

The constant multiple, sum, and difference rules are

For a product, apply the product rule:

For a quotient, apply the quotient rule:

For a composite function, apply the chain rule. If $y=f(g(x))$, then

For example,

When several rules apply, identify the structure first: sums use the sum rule, products use the product rule, quotients use the quotient rule, and nested expressions use the chain rule.

Takeaway: Accurate differentiation begins with recognizing the algebraic structure before selecting a rule.

First-Derivative Analysis

The sign of the first derivative organizes a function's behavior over intervals. If $f'(x)>0$ throughout an interval, $f$ is increasing there. If $f'(x)<0$, $f$ is decreasing. If $f'(x)=0$ throughout an interval, the function is constant.

Consider

Its derivative is

The critical numbers are $x=0$ and $x=2$. Testing the sign of $f'$ on the resulting intervals gives:

• On $(- \infty,0)$, $f'(x)>0$, so the function is increasing.

• On $(0,2)$, $f'(x)<0$, so the function is decreasing.

• On $(2,\infty)$, $f'(x)>0$, so the function is increasing.

A critical number is an input in the domain where the derivative is zero or undefined. It is a candidate for a change in behavior, not automatically an extremum.

The first derivative test classifies a critical number by its sign change:

• Positive to negative: local maximum.

• Negative to positive: local minimum.

• No sign change: neither a local maximum nor a local minimum.

For this example, the change from positive to negative at $x=0$ gives a local maximum with value $f(0)=2$. The change from negative to positive at $x=2$ gives a local minimum with value $f(2)=-2$. In contrast, $f(x)=x^3$ has $f'(x)=3x^2$, which is nonnegative on both sides of $x=0$; the critical number does not produce a local extremum.

Takeaway: Use a sign chart for $f'$ to identify increasing and decreasing intervals and to classify critical points.

Concavity and the Second Derivative

The second derivative tracks how the first derivative changes:

If $f''(x)>0$, slopes are increasing and the graph is concave up. If $f''(x)<0$, slopes are decreasing and the graph is concave down.

A possible inflection point occurs where $f''(x)=0$ or where $f''(x)$ is undefined. It is an actual inflection point only if the concavity changes there.

For

we have

Therefore, the graph is concave down for $x<1$ and concave up for $x>1$. Because the concavity changes at $x=1$, the graph has an inflection point there.

The second derivative also provides a quick test for a critical point where $f'(c)=0$:

• If $f''(c)>0$, the function has a local minimum at $x=c$.

• If $f''(c)<0$, the function has a local maximum at $x=c$.

• If $f''(c)=0$, the test is inconclusive; use the first derivative test.

Concavity and increasing or decreasing behavior answer different questions. A function can be increasing while concave down if its positive slope is becoming smaller, or decreasing while concave up if its negative slope is becoming less negative.

Takeaway: The first derivative describes the direction of change; the second derivative describes how that direction is changing.

Marginal Analysis and Optimization

Derivatives support local approximations and optimization, but conclusions must respect the domain and the model's assumptions.

If $C(q)$ is the cost of producing $q$ units, then $C'(q)$ is the marginal cost. For a small production change $\Delta q$,

Suppose

Then

At $q=100$,

The model therefore estimates an additional cost of approximately $16$ dollars for one more unit near a production level of $100$ units. This is a local approximation, not necessarily the exact cost of the next unit. The same reasoning applies to marginal revenue and marginal profit.

For global optimization on a closed interval, use a complete comparison procedure:

1. Find all critical numbers in the interior of the interval.

2. Evaluate the function at each critical number.

3. Evaluate the function at both endpoints.

4. Compare all values; the largest is the global maximum and the smallest is the global minimum.

For a rectangular garden with perimeter $40$ meters, let one side be $x$. The other side is $20-x$, so

Differentiating gives

The interior critical point satisfies

Since $A''(x)=-2<0$, the area is maximized at $x=10$. The dimensions are $10$ meters by $10$ meters, and the maximum area is $100$ square meters. Checking the endpoints confirms the global result on the stated domain.

A responsible interpretation should identify the measured quantity, state units, give the sign and magnitude, specify the relevant point or interval, distinguish exact results from approximations, and acknowledge that the conclusion describes the model rather than automatically proving that the model represents reality.

Takeaway: Optimization requires critical-point analysis, endpoint checks, domain awareness, and a clear interpretation of units and approximations.