The rate of change at a point, defined as a limit of average rates

What to assume they know

• Averaging over shorter and shorter intervals to get a speed at an instant — average rates over shrinking intervals and the chord-to-tangent picture
• Substitution works until it gives nothing over nothing, and then cancellation does — evaluating a 0/0 limit by cancelling a common factor
• Trapping a function between two others to settle the trigonometric cases — the limit of sin x over x at 0
• Limits pass through sums, products and quotients — Theorem 1
• Expanding (x + h)² and simplifying a difference of two algebraic fractions
• The tangent of the angle a line makes with the x-axis, from Equal slopes mean parallel; slopes multiplying to minus one mean perpendicular

What they should be able to do

• State Definition 1 and identify each of its parts: the point, the increment, the quotient and the limiting operation
• Explain what the phrase attached to the definition is protecting against, and give a function and a point at which the derivative does not exist
• Compute a derivative at a stated point directly from Definition 1
• Show that the difference quotient takes the 0/0 form for every function this chapter differentiates, and say why that makes §12.3.2's technique compulsory rather than optional — and note the limit of the claim: the top line shrinks to nothing only when the function has no jump at the point, which is a condition this chapter has no word for
• State Definition 2 and say how the derivative function differs from the derivative at a point
• Give the domain of a derivative function, and exhibit a function whose derivative function has a smaller domain than the function itself
• Read Fig 12.11 and identify the chord, the tangent, the increment and the angle whose tangent equals the derivative
• Write a derivative in each of the notations the chapter lists, and translate between them

Where it usually goes wrong

• "Every function has a derivative everywhere." The definition says provided the limit exists. Example 12's derivative is missing at 0, and the modulus function of §12.3 has one-sided quotients that disagree at 0, so no derivative there either. Being able to draw a function is not the same as being able to differentiate it.
• "You can just put h = 0." Then the quotient is 0/0. Every worked example in §12.5 cancels an h out of the top first, and cancelling is exactly what §12.3.2 licensed.
• "f′(a) and f′(x) are the same thing written two ways." One is a number attached to a point, the other a function. Definition 1 and Definition 2 are printed separately, two pages apart, for that reason.
• "The derivative is the tangent line." It is the slope of the tangent — a number, not a line. Fig 12.11 marks the angle ψ, and the derivative is its tangent ratio.
• "The chord becoming the tangent is a definition of tangent." In this chapter it is a reading of the algebra, not a definition; the chapter presents the geometric interpretation after the limit, not before it.
• "A constant function's derivative is 0 because there is nothing to differentiate." The chapter computes it twice, at two points and then in general, precisely so the answer rests on the definition and not on the intuition.
• "The derivative function has the same domain as the function." Example 12 is the counterexample sitting in the chapter, and the exclusion at 0 is stated explicitly for f and for f′.

Questions to check understanding

• Find the derivative of a given function at a stated point from Definition 1
• Find the derivative of a given function from first principle, leaving the point as a variable
• First-principle derivatives of a quotient such as (x + 1)/(x − 1), which require combining two fractions before cancelling
• State the definition of the derivative at a point, with the existence clause
• Given a function, state the domain of its derivative function
• Interpret f′(a) as a slope on a supplied graph, or as a rate in a stated physical situation
• Verify a stated relation between derivatives at two points, as Example 6 does

Examples worth working on the board

Values marked verified are worked out here from the chapter's printed data; no answer key was consulted.

• Definition 1 (§12.5, p. 240). f a real valued function, a a point of its domain. The derivative of f at a is the limit, as h tends to 0, of the quotient whose top is f(a + h) − f(a) and whose bottom is h — asserted only when that limit exists. Written f′(a). The chapter adds that this measures how f changes at a with respect to x.
• Example 5 (p. 240). f(x) = 3x, derivative at x = 2. Verified: the quotient is (3(2 + h) − 6)/h = 3h/h = 3 for every h ≠ 0, so the limit is 3. Note there is nothing to cancel beyond h itself, which makes this the cleanest first case.
• Example 6 (pp. 240–241). f(x) = 2x² + 3x − 5, derivatives at x = −1 and at x = 0, with the claim f′(0) + 3f′(−1) = 0 to be checked. Verified: at −1 the quotient reduces to (2h² − h)/h = 2h − 1, giving f′(−1) = −1. At 0 it reduces to (2h² + 3h)/h = 2h + 3, giving f′(0) = 3. And 3 + 3(−1) = 0.
• Example 7 (p. 241). The derivative of sin x at x = 0. Verified: the quotient is (sin h − sin 0)/h = sin h / h, whose limit is 1 by Theorem 5 (i) of §12.4. This is the first place in the chapter where the derivative machinery and the trigonometric limit meet, and it is worth pausing on: without §12.4, this derivative is unreachable.
• Example 8 (p. 241). f(x) = 3, derivative at x = 0 and at x = 3. Verified: the quotient is (3 − 3)/h = 0/h, which is 0 for every h ≠ 0, so both derivatives are 0. The chapter says first, on intuitive grounds, that a constant cannot change — and then computes anyway. Show both, in that order; that is the section's argument about what an intuition is worth.
• Fig 12.11 (p. 241), read off the printed page. The curve y = f(x) rising and bending upward. P is marked at (a, f(a)) and Q at (a + h, f(a + h)). A dashed horizontal from P and a dashed vertical from Q meet at R, which is labelled; the horizontal gap between P and R is marked h. Two straight lines are drawn through P: the steeper one passes through Q as well — the chord — and the shallower one is the tangent, whose angle with the x-axis is marked ψ. The heights f(a) and f(a + h) are marked on the vertical axis, the abscissae a and a + h on the horizontal.
• The geometric reading (p. 242). In the right triangle PQR the quotient is QR over PR, which is the tangent of the angle at P — the slope of the chord. As h tends to 0, Q slides toward P and the chord settles onto the tangent, so the derivative equals tan ψ. Verified as a consistency check: QR is f(a + h) − f(a) and PR is h, so the ratio is literally the quotient of Definition 1, not merely analogous to it.
• Definition 2 (p. 242). Taking the same limit with the point left as a variable defines a new function f′(x), and the chapter names this the first principle of derivative. Its domain is stated to be wherever that limit exists.
• The notations (p. 242). f′(x); d/dx applied to f(x); dy/dx when y = f(x); D applied to f(x); and for the value at a particular point, the d/dx form or the df/dx form carrying the point as a subscript.
• Example 9 (pp. 242–243). f(x) = 10x. Verified: the quotient is 10h/h = 10 for all h ≠ 0, so f′(x) = 10 everywhere.
• Example 10 (p. 243). f(x) = x². Verified: (x + h)² − x² = 2xh + h², and dividing by h leaves 2x + h, whose limit is 2x.
• Example 11 (p. 243). A function holding one and the same output a at every input, with a some fixed real number. Verified: the quotient is 0/h, which is 0 for h ≠ 0, so f′(x) = 0 everywhere. The chapter annotates this with the h ≠ 0 condition, which is the same cancellation licence used throughout §12.3.2.
• Example 12 (p. 243). f(x) = 1/x. Verified: the top is 1/(x + h) − 1/x = −h/(x(x + h)); dividing by h gives −1/(x(x + h)), whose limit is −1/x². Note the domain: the original function is undefined at 0 and so is its derivative, and this is the chapter's first example of a derivative function inheriting an exclusion.
• Exercise 12.2 items this topic owns (p. 248): items 1, 2, 3 ask for derivatives at a stated point — x² − 2 at x = 10, x at x = 1, 99x at x = 100 — and item 4 asks for four derivatives from first principle: x³ − 27, (x − 1)(x − 2), 1/x², and (x + 1)/(x − 1).
• Miscellaneous Exercise on Chapter 12, item 1 (p. 253) asks four more from first principle: −x, the reciprocal of −x, sin(x + 1), and cos(x − π/8).

Figures to have open

• Fig 12.11 redrawn (p. 241) at large scale with every label present: P, Q, R, the increment h, the two heights, the two abscissae, the chord, the tangent and the angle ψ. The chord must be drawn steeper than the tangent, as it is on the printed page for this upward-bending curve; getting that the wrong way round inverts the argument. Verified against the printed page.
• A movement of Q sliding down the curve toward P with the chord's inclination displayed as a number, settling on tan ψ.
• Two number lines, one for the domain of 1/x and one for the domain of its derivative, both punctured at 0.
• A notation card listing the five forms side by side. Standard schematic.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 12 "Limits and Derivatives", §12.5 Derivatives, printed pp. 239–243. Definition 1 (p. 240), Examples 5 to 8 (pp. 240–241), the geometric interpretation with Fig 12.11 (pp. 241–242), Definition 2 and the notations (p. 242), Examples 9 to 12 (pp. 242–243).
• Exercise 12.2 items 1–4, printed p. 248.
• Miscellaneous Exercise on Chapter 12, item 1, printed p. 253.
• Chapter Summary, p. 255, which prints both definitions.
• Printed stale cross-reference, confirmed on the page image. The opening sentence of §12.5 on p. 239 sends the reader to "Section 13.2" for the position-and-rate discussion. That material is §12.2 of this book, pp. 217–220 — a leftover from the pre-2022 edition, in which this was Chapter 13. Cite §12.2.