Why would differentiating the substituted area give zero?

You made it constant. Substitution first removes the variable whose change you need.

For negative inputs moving towards zero, what happens to the positive squared outputs?

They decrease. The derivative is negative throughout that interval.

Sine rises near zero; what does it do between pi/2 and pi?

It decreases. Its derivative, cosine, is negative throughout that part of the interval.

Here, what does zero identify: the minimum output or its input?

Its input. Putting zero into the function gives output 1.

If slopes stay positive on both sides of a zero slope, which extremum occurs?

Neither. The function keeps rising instead of changing direction.

When both derivatives vanish, which first-derivative sign change would prove a local minimum?

Negative to positive. The function falls before the point and rises after it.

Which inputs supply both absolute extremes in this example?

The endpoints, −1 and 1. Their outputs lie below and above the interior candidate's output.

With this boundary fixed, what must happen to the width when the length increases?

The width must decrease. The fixed perimeter links the two dimensions.

Teach Application of Derivatives in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

A speedometer shows how fast a vehicle moves at that moment.
A rising temperature can still be below zero.
Boxes made from the same sheet can hold different amounts.

What to teach

How to teach it

Rates and direction of change

$A=\pi r^2$

$\frac{dA}{dr}=2\pi r$

$\left.\frac{dA}{dr}\right|_{r=1}=2\pi\times1$

1. Rate of Change of Quantities

A derivative measures how quickly one quantity changes against another.

Show: Write these lines for a ripple. Name area A, radius r, and the circle constant pi.
Ask: At radius 1 cm, how quickly do you predict area changes with radius at that instant?
Say: The rate is two pi square centimetres per centimetre.
Used in: Water-level rates help estimate when a tank will overflow.

If your child says…“I'll put the radius in first, then differentiate.”

Ask: Does replacing the changing radius with one number preserve how the area changes?
What’s true: Differentiate before inserting the current radius; a shrinking radius has a negative time rate.

Parabola with tangents showing negative, zero and positive slopes from left to right.

2. Increasing and Decreasing Functions

Slope signs show where outputs rise or fall as inputs increase.

Show: Follow the diagram's parabola left to right. Call f prime the slope.
Ask: Where do these slopes predict falling and rising outputs?
Say: Negative slopes mean falling outputs; positive slopes mean rising outputs. Check intervals separately wherever the domain has gaps.
Used in: Sales trends show when demand rises, helping shops plan stock.

If your child says…“The square is positive, so the function increases.”

Ask: As a negative input approaches zero, does its square grow or shrink?
What’s true: Direction depends on the slope, not whether the output is positive.

Aligned sine and cosine graphs connect sine's peak with cosine's change of sign.

3. Increasing and Decreasing Functions

Check slope signs throughout the allowed interval, including any parameter's effect.

Show: Look at the diagram, with angles in radians. Follow cosine below from zero to pi. Follow the dashed guide up to sine.
Ask: Where does cosine's sign predict that sine will change from rising to falling?
Say: Sine turns at $\frac{\pi}{2}$ because its derivative, cosine, changes from positive to negative. Cosine controls the direction here.
Used in: Calculations for a swing predict when its direction reverses.

If your child says…“Sine and cosine stay positive throughout any interval.”

Ask: Where does the cosine curve lie below its horizontal axis?
What’s true: Cosine becomes negative after half pi within this interval.

Finding greatest and least values

$f(x)=x^2+1$

$f(x)-1=x^2$

$f(0)=0^2+1$

4. Maxima and Minima

Greatest and least outputs must actually occur within the domain.

Show: Write these lines for x squared plus one, allowing every real input. Say: A square cannot be negative.
Ask: Before evaluating, what do you predict the smallest output will be?
Say: The minimum value is 1, reached at input zero. There is no maximum because squares grow without bound.
Used in: Capacity limits tell packers how much a container can hold.

If your child says…“The minimum value is zero because x is zero.”

Ask: Which number does the function produce when you put zero in?
What’s true: The input locates the minimum; the output gives its value.

Two parabolas show opposite derivative sign changes at a local maximum and minimum.

5. Local Maxima and Minima; First Derivative Test

Nearby slope changes distinguish local peaks from local dips.

Show: Read the diagram's slope signs left to right beneath each curve.
Ask: Which curve has a local maximum at zero?
Say: Rising then falling gives the left curve a local maximum. Falling then rising gives the right curve a local minimum.
Used in: Road profiles locate nearby crests and dips, helping plan drainage.

If your child says…“Every zero slope must give a maximum or minimum.”

Ask: What do the slopes on either side tell us that zero alone cannot?
What’s true: A zero slope alone does not establish either kind of extremum.

$f(x)=(x-1)^2$

$f'(x)=2(x-1)$

$f''(1)=2$

$f(1)=(1-1)^2$

6. Second Derivative Test

At zero slope, slope change can identify a peak or dip.

Show: Write these lines. Name f double prime the second derivative: it measures how slope changes. Point out that the first derivative vanishes at input 1.
Ask: Before evaluating, what minimum output do you predict?
Say: The minimum is zero: the positive second derivative shows the slope rising through zero.
Used in: Cost models use minimum tests to identify cheaper production levels.

If your child says…“A zero second derivative means there cannot be a minimum.”

Ask: If slopes still change from negative to positive, what does that show?
What’s true: A zero second derivative leaves this test undecided.

$f'(x)=3x^2$

$f(-1)=-1$

$f(0)=0$

$f(1)=1$

7. Maximum and Minimum Values of a Function in a Closed Interval

Compare all critical-point outputs and both endpoint outputs.

Show: Write the derivative first for x cubed, from −1 to 1, including both endpoints. After the prediction, write and compare the candidate outputs.
Ask: Before evaluating the candidates, where do you predict the greatest and least outputs occur?
Say: The minimum is −1 and the maximum is 1; both occur at endpoints.
Used in: Checking a day's full temperature record finds its hottest and coldest readings.

If your child says…“Only the zero-slope point needs checking.”

Ask: Which listed outputs are smaller or larger than the output at zero?
What’s true: An endpoint can beat every interior critical point.

Choosing the best dimensions

$y=12-x$

$A(x)=x(12-x)$

$A'(x)=12-2x$

$A'(x)=0\ \Rightarrow\ x=6$

8. Simple Problems on Maxima and Minima

Keep the constraint, then find and verify the best allowed value.

Show: Write these lines for rectangle sides x and y, perimeter 24 cm, and area A.
Ask: What greatest area do these lines predict?
Say: The 6 cm square gives 36 square centimetres. Across allowed sides, the derivative changes from positive to negative at 6, proving the maximum.
Used in: Package designers maximise capacity within material limits, reducing waste.

If your child says…“Both sides can keep growing while the boundary stays fixed.”

Ask: If both sides grew, what would happen to their fixed sum?
What’s true: With a fixed perimeter, lengthening one side forces the other to shorten.

A real problem

Kepler studied how to measure wine barrels.

His 1615 work examined wine-barrel measurement.

The work was called Stereometria Doliorum.

His work confirmed that a simple measuring rod worked for Austrian wine casks.

University of St Andrews Dictionary of Scientific Biography, hosted by St Andrews

Wooden barrel with a measuring rod inserted through its side opening.

Concept Lab · Common misconception

The tempting mistake in Application of Derivatives

Open standalone misconception page →

Does a zero derivative guarantee a maximum or minimum?

Wrong

Every point where f′(x) = 0 is a turning point.

Why it feels right. Smooth maxima and minima often have horizontal tangents.

Right

A stationary point can have a horizontal tangent while the function keeps increasing through it.

Worked example

The curve flattens at zero but rises on both sides. There is no local maximum or minimum there.

For f(x) = x³, f′(0) = 0.
Values go from −1 at x = −1 to 0 to 1 at x = 1, and the function increases throughout.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

Why would differentiating the substituted area give zero?

You made it constant. Substitution first removes the variable whose change you need.
For negative inputs moving towards zero, what happens to the positive squared outputs?

They decrease. The derivative is negative throughout that interval.
Sine rises near zero; what does it do between $\frac{\pi}{2}$ and $\pi$ ?

It decreases. Its derivative, cosine, is negative throughout that part of the interval.
Here, what does zero identify: the minimum output or its input?

Its input. Putting zero into the function gives output 1.
If slopes stay positive on both sides of a zero slope, which extremum occurs?

Neither. The function keeps rising instead of changing direction.
When both derivatives vanish, which first-derivative sign change would prove a local minimum?

Negative to positive. The function falls before the point and rises after it.
Which inputs supply both absolute extremes in this example?

The endpoints, −1 and 1. Their outputs lie below and above the interior candidate's output.
With this boundary fixed, what must happen to the width when the length increases?

The width must decrease. The fixed perimeter links the two dimensions.

15 more questions with answers, 1 with pictures

Teach Application of Derivatives in eight steps.

Why it matters