Math Class Didn’t Show Its Work, companion reader · Chapter 29

Rules of Change, Unmasked

A derivative rule is a receipt for work someone has already checked.

These four demonstrations open the receipts. Each rule is set beside the slow calculation it replaces, with the chapter's own functions, so you can see the shortcut and the check agree, or catch the place where a shortcut goes wrong.

Every example in these readers is a constructed teaching example taken from or modeled on the book's own worked examples. The numbers are declared inputs chosen to make the mathematics visible. They are not measurements of any real person, product or study.

Demonstration 1 of 4

Constants have no change to report

Adding 700 to x^2 lifts the whole graph. Does it change the slope anywhere?

Both curves share one vertical scale (broken for 700). The shifted curve is the same curve moved straight up, so its tangent at the dot is parallel to the original.

\[d(x^2)/dx = 2x\]

\[d(x^2 + 700)/dx = 2x\]

d/dx means the rate of change as x changes. c is a constant: it stays fixed while x moves, so its own rate is 0.

Predict first. Move the point to x = 3. What slope do both panels show there?

Choose an example

Figure: Constants have no change to report. y = x squared and y = x squared plus 700 near x = 1, drawn at the same scale with the vertical axis broken to skip the gap. Both have dashed tangents of slope 2 at x = 1, so the tangents are parallel.
Added constant c: 700, Point x: 1
Constructed example: the chapter's comparison of x^2 and x^2 + 700, with other constants for contrast.

Calculated values

Constant c
700
Height of x^2 at x = 1
1^2 = 1
Height of x^2 + 700 at x = 1
1^2 + 700 = 701
Slope of x^2
2(1) = 2
Slope of x^2 + 700
2(1) + 0 = 2

At x = 1 the heights are 1^2 = 1 and 1^2 + 700 = 701, but both slopes are 2 x 1 = 2. The constant 700 adds a rate of 0: (c - c)/h = 0/h = 0. It moves the graph up without tilting it.

Use the idea

When a formula has a fixed added amount, such as a flat fee, the fee changes the total but not how fast the total changes.

Where the conclusion applies

c must be constant with respect to x. If the added amount itself depended on x, it would contribute its own rate.

Check your understanding: What is the slope of x^2 + 700 at x = 5?
2(5) + 0 = 10, the same as the slope of x^2 at x = 5.

Chapter 29 source: section "Constants have no change to report". Demonstration C29-D01.

Demonstration 2 of 4

The power pattern from the quotients

At x = 2, which of x, x^2 and x^3 changes fastest, and where does n x^(n - 1) come from?

For each power the quotient simplifies to the rule's value plus terms that still contain h. Those terms fade as h shrinks, so each curve heads to its open circle: 1, 4 and 12.

\[= 3x^2 + 3xh + h^2\]

\[d(x^n)/dx = n x^{(n - 1)}\]

h is a nonzero step from x = 2. The quotient [(2 + h)^n - 2^n]/h is the average rate over that step. n is a positive whole-number power.

Predict first. Pick x^2 and h = 0.1. Predict the quotient before you look, using 2x + h.

Choose an example

Figure: The power pattern from the quotients. Three curves of difference quotients at x = 2 for x, x squared and x cubed, with open circles at h = 0 at heights 1, 4 and 12. The x^3 curve is bold and its point at h = 0.1 has height 12.61.
Function: x^3, Step h: 0.1
Constructed example: the chapter's starting question comparing x, x^2 and x^3 at x = 2.

Calculated values

Function
x^3
Change over the step
2.1^3 - 2^3 = 1.261
Quotient
1.261/0.1 = 12.61
Simplified quotient
3x^2 + 3xh + h^2
Power rule at x = 2
3(2^2) = 12

For x^3 at x = 2 with h = 0.1: (2.1^3 - 2^3)/0.1 = 1.261/0.1 = 12.61. The simplified quotient 3x^2 + 3xh + h^2 gives 3(4) + 3(2)(0.1) + 0.1^2 = 12.61. As h shrinks the leftover h terms vanish, leaving 3 x 2^2 = 12, the power rule's value.

Use the idea

Use the power rule as a stored result, and spot check it with one quotient at a small h when an answer looks surprising.

Where the conclusion applies

Positive whole-number powers only, as the chapter states the rule. The ranking at x = 2 need not hold at every x.

Check your understanding: For x^3 at x = 2 with h = 1, what is the quotient?
(3^3 - 2^3)/1 = 27 - 8 = 19, and 3(4) + 3(2)(1) + 1 = 19.

Chapter 29 source: section "The power pattern". Demonstration C29-D02.

Demonstration 3 of 4

A product is not two isolated rates

When both factors of p(x) = x^2(x + 1) change, where does the change in the product come from?

The area grows by two strips and a corner. Divided by h, the strips head to f'g and fg', while the corner, a change times a change, fades away.

\[d[fg]/dx = f'g + fg'\]

\[p(x) = x^2(x + 1)\]

f = x^2 is the width and g = x + 1 the height, so the area is fg. Delta f and Delta g are their changes when x grows by h; here Delta g = h.

Predict first. Shrink h from 0.5 to 0.1. Which piece shrinks fastest compared with h: a strip or the corner?

Choose an example

Figure: A product is not two isolated rates. A rectangle of width 1 and height 2 grown by 1.25 to the right and 0.5 upward. A right strip of area 2.5, a top strip of area 0.5 and a small corner of area 0.625 are shaded separately.
Starting x: 1, Step h: 0.5
Constructed example: the chapter's p(x) = x^2(x + 1) drawn as a changing rectangle.

Calculated values

f and g at x
1 and 2
Delta f
1.5^2 - 1^2 = 1.25
Delta g
0.5
Area change
2.5 + 0.5 + 0.625 = 3.625
Area change / h
3.625/0.5 = 7.25
Product rule p'(x)
(2x)(x + 1) + x^2(1) = 4 + 1 = 5
Corner share / h
0.625/0.5 = 1.25

At x = 1 with h = 0.5: the strips are 2 x 1.25 = 2.5 and 1 x 0.5 = 0.5, and the corner is 1.25 x 0.5 = 0.625. Total 3.625, so the average rate is 3.625/0.5 = 7.25. The corner contributes 1.25 of that and shrinks away as h shrinks; the strips head to 2x(x + 1) + x^2 = 5, never to f'g' = 2 x 1 = 2.

Use the idea

When two changing quantities multiply, such as a price and a quantity, count both strips: each factor's change times the other factor's current size.

Where the conclusion applies

Both factors differentiable, as in the chapter. The picture motivates the rule; the theorem proves it. It is f'g + fg', never f'g'.

Check your understanding: Using the product rule, what is p'(1) for p(x) = x^2(x + 1)?
(2 x 1)(1 + 1) + 1^2(1) = 4 + 1 = 5, matching 3(1)^2 + 2(1) = 5.

Chapter 29 source: section "A product is not two isolated rates". Demonstration C29-D03.

Demonstration 4 of 4

Check the chain rule at one point

Does the chain rule's 6 for q(t) = (3t + 1)^2 at t = 0 match the slow quotients?

For every nonzero h the quotient is 6 + 9h, a line with a hole at h = 0. The rule's claim is the dashed line; a correct rule meets the line at the hole.

\[q'(0) = 6(1) = 6\]

\[= 6 + 9h\]

The inside is u = 3t + 1 with rate 3; the outside squares u with rate 2u. At t = 0, u = 1. h is a nonzero step from t = 0.

Predict first. Switch to the rule that forgets the inside rate. How far is its answer from the quotients?

Choose an example

Figure: Check the chain rule at one point. A line of average rates 6 + 9h for h from -0.2 to 0.2, with an open circle at h = 0, height 6. A dashed line marks the claimed rate 6. The point h = 0.01 has height 6.09.
Step h: 0.01, Rule being checked: Chain rule
Constructed example: the chapter's nested square q(t) = (3t + 1)^2 and its steps h = 0.01 and -0.01.

Calculated values

q(h)
(3(0.01) + 1)^2 = 1.0609
Difference quotient
(1.0609 - 1)/0.01 = 6.09
Simplified
6 + 9(0.01) = 6.09
Rule's claimed q'(0)
2(1)(3) = 6
Agrees with the quotients?
yes, they approach 6

With h = 0.01: [(3(0.01) + 1)^2 - 1]/0.01 = (1.0609 - 1)/0.01 = 6.09, matching 6 + 9(0.01) = 6.09. Quotients from the right sit above 6 and from the left below it, closing in on the chain rule's 6, so the rule passes this spot check.

Use the idea

After using a rule, test it at one point with a small positive and a small negative step. A missing factor shows up at once.

Where the conclusion applies

One spot check at one input. It can expose a wrong rule but does not prove the chain rule everywhere.

Check your understanding: What is the quotient at h = 0.1?
6 + 9(0.1) = 6.9. Check: (1.3^2 - 1)/0.1 = 0.69/0.1 = 6.9.

Chapter 29 source: section "Check the chain rule at one point". Demonstration C29-D04.