Rules for differentiating a sum, a product and a quotient

What to assume they know

• The rate of change at a point, defined as a limit of average rates — Definition 2 and computing from first principle
• Limits pass through sums, products and quotients — Theorem 1, whose four parts these mirror
• Algebraic manipulation of products and quotients of expressions
• Trapping a function between two others to settle the trigonometric cases — the trigonometric limits, needed only for the worked examples that involve sin and cos
• The Pythagorean identity relating sin², cos² and 1

What they should be able to do

• State each of the four parts of the derivative theorem in §12.5.1 in your own words, including the condition attached to the quotient part
• Write the product and quotient rules in the compressed u, v notation the chapter supplies, and translate between that and the full notation
• Explain why the derivative of a product is not the product of the derivatives, and produce a two-line counterexample
• Differentiate one function by two different decompositions and show the answers agree
• Apply the quotient rule to a rational expression and state where the result fails to be defined
• Apply the product rule to a function written as a square, and recognise when a trigonometric identity gives a shorter route to the same answer
• Identify which rule is doing the work at each step of a multi-rule computation
• Derive the derivative of cot x by two routes and check that they agree

Where it usually goes wrong

• "The derivative of a product is the product of the derivatives." Section 4's counterexample kills it in one line with the simplest possible function. Do this before stating the correct rule, not after.
• "The two terms of the product rule are interchangeable, so the order does not matter." In the product rule it genuinely does not — the two terms are added. In the quotient rule it does, because the numerator is a difference and swapping the terms flips the sign of the whole answer. Students carry the product rule's forgiveness into the quotient rule.
• "The quotient rule's denominator is the derivative of the lower function squared." It is the lower function itself, squared, undifferentiated.
• "If two decompositions give different answers, one of them is a different function." They cannot give different answers, and 10x done two ways on p. 245 is the chapter's demonstration of that. A mismatch means an arithmetic slip, and checking by a second decomposition is a real technique.
• "Every function needs the rules; first principle is only for the syllabus." Some do not — the derivative of x, on p. 245, is got from the definition in one line, and everything else in the section is built on it.
• "The chapter proves these rules from the limit theorem." It states that they follow from it and explicitly declines to prove them. Do not put a proof in a student's mouth as though the book supplied it.
• "sin²x and sin 2x differentiate the same way because Example 18's answer is sin 2x." Example 18's answer is the derivative of sin²x, which happens to equal sin 2x. Example 21 (i) differentiates sin 2x itself and gets something else entirely. Put the two side by side.

Questions to check understanding

• Differentiate a product of two polynomials by the product rule
• Differentiate a rational function by the quotient rule and state where the derivative is undefined
• Differentiate a trigonometric product such as sin x cos x, and a reciprocal such as sec x or cosec x, by the quotient rule
• Differentiate one function by two routes and confirm the answers agree
• State the product and quotient rules in the u, v notation
• One-mark: give a pair of functions showing that the derivative of a product is not the product of the derivatives
• Longer items combining the rules with the power rule, as throughout the Miscellaneous Exercise

Examples worth working on the board

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

• The theorem (§12.5.1, p. 244). Two functions whose derivatives are defined on a shared domain. Four claims: (i) the derivative of the sum is the sum of the derivatives; (ii) the same for the difference; (iii) the derivative of the product is the first differentiated times the second, plus the first times the second differentiated; (iv) the derivative of the quotient has, on top, the lower function times the derivative of the upper minus the upper times the derivative of the lower, and underneath, the square of the lower function — subject to the lower function not being zero. The chapter states that these follow from the corresponding limit theorem and that it will not prove them here. This theorem is printed as Theorem 5, which is also the number carried by the trigonometric-limits theorem on p. 235. Both labels are printed; confirmed on the page images. Cite by page.
• The compressed forms (pp. 244–245). Putting u for the first function and v for the second, the product rule reads (uv)′ = u′v + uv′ and the quotient rule reads (u/v)′ = (u′v − uv′)/v². The chapter attaches Leibnitz's name to the product form.
• The derivative of x (p. 245). Verified from the definition: the quotient is ((x + h) − x)/h = h/h = 1 for h ≠ 0, so the derivative is the constant function 1. Everything in this section is built on this one line plus the derivative of a constant being 0.
• 10x by the sum rule (p. 245). Writing 10x as x added to itself ten times and applying part (i) ten times gives ten copies of 1, so 10. Verified.
• 10x by the product rule (p. 245). Writing 10x as u times v with u the constant 10 and v(x) = x: u′ = 0 and v′ = 1, so u′v + uv′ = 0·x + 10·1 = 10. Verified, and it agrees. Run these two side by side — this is the section's test that the rules are consistent, and it is the chapter's own pairing.
• x² by the product rule (p. 245). Writing x² as x·x gives 1·x + x·1 = 2x, which is what Example 10 (p. 243) got from first principle. Verified.
• The naive-rule counterexample (section 4; not in the book, not printed). Take both functions to be x. The product is x², whose derivative is 2x. The product of the derivatives is 1 × 1 = 1. These differ at every x other than 1/2, so the naive rule is not merely imprecise, it is false.
• Example 15 (p. 247). The function (x + 1)/x, handled by the quotient rule. Take the upper function to be x + 1 and the lower to be x; each of them differentiates to 1. Verified: the rule's numerator becomes x − (x + 1), which is −1, so the derivative is −1/x². The chapter notes at the start that this function has no value at 0, and the derivative inherits that.
• Example 18 (p. 248). The derivative of sin²x by the product rule, taking both factors to be sin x. Verified: the rule's two terms come out identical here, each a sine multiplied by a cosine, so the total is twice that product — which the chapter then rewrites, by the double-angle identity, as sin 2x.
• Example 21 (i) (p. 251). The derivative of sin 2x, obtained by first rewriting it as 2 sin x cos x and then applying the product rule. Verified: differentiating the sine factor contributes a squared cosine, differentiating the cosine factor contributes a squared sine with a minus in front, and the leading 2 carries through, giving 2(cos²x − sin²x). Set this beside Example 18: the same identity is used in opposite directions in the two examples, which is worth pointing out.
• Example 21 (ii) (pp. 251–252). cot x, twice. Route one: quotient rule on cos x over sin x. Verified: the rule's numerator comes out as minus the sum of the two squares, which the Pythagorean identity collapses to −1, so the whole thing is −1/sin²x, written −cosec²x. Route two: quotient rule on 1 over tan x, using the tangent derivative established at Example 17 — a squared secant — together with the fact that a constant differentiates to 0. Verified: only the second term of the numerator survives, leaving a squared secant with a minus in front, divided by a squared tangent; that simplifies to the same −1/sin²x. Both routes are printed; run both, because the agreement is the point.
• Example 22 (pp. 252–253). Two quotient-rule computations left partly unsimplified in the book: (x⁵ − cos x)/sin x and (x + cos x)/tan x. The first is worked to ((5x⁴ + sin x) sin x − (x⁵ − cos x) cos x)/sin²x, which the chapter then expands; the second is left in the assembled form.
• Exercise 12.2 items this topic owns (p. 249): item 7 (three products and quotients built from constants a and b), item 9 (six functions, several needing the product or quotient rule), item 11 parts (i) to (vii) (trigonometric products, secants, cosecants and combinations).
• Miscellaneous Exercise on Chapter 12 (pp. 253–254): items 3 to 30 are almost all product-rule and quotient-rule practice, including items 16 to 21 and 23 to 30 where a trigonometric function sits in one or both positions.

Figures to have open

• A four-row comparison card: limit rule on the left, derivative rule on the right, with rows three and four marked as not parallel. Standard schematic; the chapter prints no figure in §12.5.1, which I confirmed on p. 244.
• A colour-tracked movement of the product rule, in which each term of the answer lights up the factor that was differentiated to produce it.
• A two-column agreement panel used three times — 10x, x², cot x — with the two routes converging.
• No textbook figure is needed for this topic.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 12 "Limits and Derivatives", §12.5.1 Algebra of derivative of functions, printed pp. 244–245, including the compressed u, v forms and the Leibnitz naming.
• Worked applications: Example 15 (p. 247), Example 18 (p. 248), Examples 21 and 22 (pp. 251–253).
• Exercise 12.2 items 7, 9, 11, printed p. 249; Miscellaneous Exercise on Chapter 12, items 3–30, pp. 253–254.
• Chapter Summary, p. 255, which prints the three compressed forms together.
• Backward reference inside the chapter: the corresponding limit theorem is Theorem 1, §12.3.1, p. 228.