Reading a derivative as how fast one quantity answers another

What to assume they know

• Differentiating a polynomial, and the derivative of a power, from Chapter 5
• The two notations for a derivative — the ratio form and the primed form
• Evaluating a derivative at a stated input
• Area and circumference of a circle; volume and surface area of a cube
• Units, and how a ratio of two units is read aloud
• Function notation, and what it means to declare one quantity a function of another
• Rounding a money value to two decimal places

What they should be able to do

• Read a derivative aloud as a statement about two named quantities rather than as a computation
• Write both the derivative as a function and its value at one stated input, in the notations the chapter uses for each
• Compute the rate of change of a geometric quantity with respect to a length, and state the units that the computation actually produces
• Detect a units mismatch between a question, its working and its printed answer
• Say what the sign of a derivative reports about the two quantities, and cite the chapter's own note for it
• Recognise marginal cost and marginal revenue as instantaneous rates rather than as differences between successive outputs
• Evaluate a marginal quantity at a stated output level and round a money answer correctly
• Distinguish a rate taken with respect to a length from a rate taken with respect to time, and say which of the two a given question is asking for
• Name the two subjects §6.1 announces that no section of the chapter goes on to teach, and avoid repeating the announcement

Where it usually goes wrong

• "A rate always means per second." It means per unit of whatever the derivative was taken against. Example 1 and Exercise 6.1 Q9 and Q13 are all rates against a length. A student who hears every derivative as a speed will reach for a time variable that the question never supplied.
• "The derivative and its value at a point are the same thing." One is a function and the other is a number, and §6.2 sets out separate notation for each in its first paragraph. Asked for a rate at a stated input, a student who hands back the function has answered a different question.
• "Marginal cost is the cost of the next unit." It is the derivative evaluated at the current output — a rate at an instant, as both Examples 5 and 6 say in their own statements. The difference between successive costs is a close cousin and not the same number, and the chapter's definition is the rate.
• "The units look after themselves." They do not, and Example 1 is the proof: its printed answer carries a time unit that its working never introduces.
• "Rates are about motion." Five of the six worked examples in §6.2 are, and the two money examples are not. Cost against output has no time in it at all, which is exactly why the chapter puts them last.
• "This chapter will teach me tangents and normals, because the introduction says so." It says so and does not. Nothing under either name is taught in Part I pp. 147–186. Say what the chapter actually covers and move on.
• "A negative rate means the quantity is negative." It means the quantity is falling as the other rises. The chapter's boxed note on Part I p. 149 says exactly this, and it is the single sentence §6.3 will later promote to a theorem.
• "Pi has to be turned into a decimal." Every answer in Exercise 6.1 that involves a circle or a sphere is cleanest left as a multiple of pi, and the chapter leaves them that way. Decimalising early loses the pattern the exercise is drilling.

Questions to check understanding

• Differentiate a stated area or volume formula with respect to a length and evaluate at a given value, giving the units
• Given a question, a worked solution and a printed answer, say whether the three agree about which quantity the rate is taken against
• Write the derivative of a stated function and, separately, its value at a stated input, using the correct notation for each
• Compute a marginal cost or marginal revenue at a stated output level from a printed cost or revenue function
• State what a negative rate says about the two quantities involved
• Choose the correct value of a rate from four options — the form of Exercise 6.1 Q17 and Q18
• Given a rate and its units, name the two quantities it relates

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its own exercises.

• The §6.1 opening paragraph (Part I p. 147). It looks back at Chapter 5, says the chapter will apply derivatives across several fields, and then lists what is coming as three numbered items plus a closing sentence. Items one and three land — rates of change, and turning points leading to largest and smallest values — and so does the sentence about intervals of rise and fall. Item two and the closing sentence do not. See Notes.
• The distance-and-time sentence (§6.2 opening, Part I p. 147). The chapter starts from the one rate a student already reads without translation, then replaces distance by any quantity and time by any other. Use this as the whole argument of sections 3 and 4: nothing new is being defined, an old reading is being widened.
• The two notations (§6.2, Part I p. 147). The chapter sets out the derivative as a function of the input, and separately the same derivative carrying a vertical bar and a subscript that fixes the input at one value; it gives the primed form of each alongside. Read off the printed page, the subscripted form is set with the bar to the right of the ratio. Section 5 exists because students collapse the two and then report a function where a number was asked for.
• Example 1 (Part I p. 148). The area of a circle against its radius. The chapter takes the derivative of pi r squared with respect to r, reaches two pi r, and puts r equal to five. Verified: the value is ten pi. The units are the teaching point. The derivative taken is with respect to a length, so its units are area per length — square centimetres per centimetre, which is centimetres. The question's phrase fixes seconds as the second quantity and the printed answer carries square centimetres per second, but no time derivative is taken anywhere in the working. Recorded in Notes.
• The Note on the sign (Part I p. 149). One boxed sentence: the rate is positive when the two quantities move the same way and negative when they move opposite ways. It is the only interpretive remark in §6.2 and it is worth thirty seconds on its own, because §6.3 turns exactly this sentence into a theorem.
• Example 5 (Part I p. 150). Total cost as a cubic in the number of units, with a definition of marginal cost supplied inside the question: how sharply the total cost answers a shift in output. The chapter differentiates and evaluates at three units. Verified: the derivative is fifteen thousandths of x squared, less four hundredths of x, plus thirty; at three units that is one hundred thirty-five thousandths, less twelve hundredths, plus thirty, which is thirty point zero one five. The chapter reports it as a money value rounded to two places. Note: the definition of the term is given in the question, not in the exposition.
• Example 6 (Part I p. 150). Total revenue as a quadratic in the number sold; marginal revenue defined in the question as before. Verified: the derivative is six x plus thirty-six, and at five units it is sixty-six. This is the cleanest instance in §6.2 of a rate whose answer is a plain number with a currency attached and no compound unit at all, which is why it belongs beside Example 1 rather than after it.
• Exercise 6.1 Q1 (Part I p. 150). The same circle as Example 1, asked twice without the phrase about seconds and without units. Verified: six pi at radius three, eight pi at radius four. Put this next to Example 1 — the exercise asks the question the example meant to ask.
• Exercise 6.1 Q9 and Q13 (Part I p. 151). Two spheres. Q9 asks for the rate at which volume grows with the radius at radius ten. Verified: the derivative of four thirds pi r cubed is four pi r squared, which is four hundred pi. Q13 gives the diameter as three halves of one more than twice x and asks for the rate of change of volume with respect to x. Verified: the radius is three quarters of that same bracket, the volume is nine sixteenths pi times the bracket cubed, and the derivative is twenty-seven eighths pi times the bracket squared. Both are rates against a length, not against time, and both belong to this topic rather than the next.
• Exercise 6.1 Q15 and Q16 (Part I p. 151). The marginal-cost and marginal-revenue drills. Verified: Q15's derivative is twenty-one thousandths of x squared, less six thousandths of x, plus fifteen; at seventeen units it is twenty point nine six seven. Q16's derivative is twenty-six x plus twenty-six; at seven units it is two hundred and eight.
• Exercise 6.1 Q17 and Q18 (Part I pp. 151–152). The exercise's two multiple-choice items, introduced by a line naming both. Verified: Q17 asks the circle's area rate at radius six, which is twelve pi, option B. Q18 reuses Example 6's revenue function at fifteen units, giving one hundred twenty-six, option D. Q18 is worth showing because it is Example 6 with one number changed, and a student who has understood the example can answer it without writing anything.
• The chapter's opening page (Part I p. 147). It carries a chapter number in a coloured box, a boxed title, an epigraph attributed to Whitehead about calculus as a key, and a QR code whose printed label combines the Part I catalogue number with the chapter number. Caption these on a card; do not reproduce the page. See Notes about the epigraph's punctuation.
• The Summary's two rate bullets (Part I p. 185). The first restates §6.2's two notations, the second restates the Chain Rule link. Both are faithful to §6.2. Use them as the end card for this topic.

Figures to have open

• A strip diagram for section 3 showing distance accumulating against time, with the ratio of two small increments picked out. An added device; §6.2 prints no figure at all.
• A two-row notation card for section 5, the general derivative above and the derivative at a fixed input below. The content is the chapter's own Part I p. 147 display; the layout is added here.
• A unit-cancellation strip for sections 6 and 7: square centimetres over centimetres reducing to centimetres, set beside the printed answer's square centimetres over seconds. This is the load-bearing figure of the topic and it is entirely added here; the chapter draws nothing here.
• Two small arrow pairs for section 8, one with both arrows up and one with arrows opposed. Not in the book; the chapter's note is prose only.
• A caption card for the chapter opener (Part I p. 147) naming the epigraph and its attribution. Ignore the QR code. Do not reproduce the printed epigraph line — see Notes.

Where this sits in the book

• NCERT Class 12 Mathematics, Part I, Chapter 6 "Application of Derivatives", §6.1 Introduction, Part I p. 147
• §6.2 Rate of Change of Quantities, the two notations and the Chain Rule link, Part I p. 147
• Example 1, Part I p. 148; the boxed note on the sign of a rate, Part I p. 149
• Examples 5 and 6, Part I p. 150
• Exercise 6.1, questions 1, 9, 13, 15, 16, 17 and 18, Part I pp. 150–152
• Summary, the first two bullets, Part I p. 185