The power rule, and a polynomial's derivative assembled out of it and the sum rule

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
• Rules for differentiating a sum, a product and a quotient — the sum and product rules
• The binomial expansion of a shifted power, from A rule that turns a position number into a term
• Proof by induction on a whole number
• Summing the first n whole numbers
• Laws of exponents, including negative exponents

What they should be able to do

• State the power rule for a positive whole exponent
• Prove it by binomial expansion, identifying which factor makes every term beyond the first vanish in the limit
• Prove it again by induction, naming the base case and the rule used in the inductive step
• State what the chapter's Remark extends the rule to, and note that no proof of the extension is given here
• Differentiate a polynomial by combining the power rule with the sum rule, and identify which rule licenses each step
• Evaluate a polynomial derivative at a stated point, including cases where the arithmetic needs a summation formula
• Apply the power rule to negative exponents and to expressions rewritten to make the exponent visible
• Recognise when a product or quotient can be expanded into a polynomial first, making the power rule the shorter route

Where it usually goes wrong

• "The power rule works because you bring the exponent down." That is the recipe, not the reason. The reason is that the second binomial term is the only one that survives division by the increment without still carrying a factor of it, and its coefficient is n.
• "The induction proof and the binomial proof are the same argument." They share nothing but the conclusion. One needs the binomial theorem; the other needs the product rule and no expansion at all. Show both and say which equipment each consumes.
• "Theorem 7 is a new formula to learn." It is two results already proved, used in sequence. The chapter's one-sentence proof is the giveaway.
• "The rule needs a whole-number exponent." Theorem 6 is stated for positive whole numbers, and the Remark widens it to any real exponent without proof. Exercise 12.2 q9 (iii), (iv) and (v) cannot be done inside the theorem as stated; they need the Remark.
• "A constant term differentiates to itself." It differentiates to 0, which was settled at Example 11 on p. 243. The constant term of a polynomial simply disappears.
• "You must use the product rule on x⁻³(5 + 3x)." You may, but expanding into a sum of two powers first is shorter and is what the exercise is set up for. Doing it both ways is a useful check.
• "Example 14 needs you to write out fifty terms." It needs the summation formula for the first fifty whole numbers. The bookkeeping is the point of the question.

Questions to check understanding

• Differentiate a polynomial and evaluate the result at a stated point
• Prove the power rule from first principle using the binomial expansion
• Prove the power rule by induction, naming the rule used in the step
• Differentiate an expression with negative exponents after expanding it
• Prove a stated relation between a polynomial's derivatives at two points
• One-mark: state the derivative of a constant, and of x itself
• Differentiate a bracketed product by expanding first, and confirm the answer by the product rule

Examples worth working on the board

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

• Theorem 6 (§12.5, p. 245). For any positive whole number n, the derivative of the nth power of x is n times the (n − 1)th power of x.
• Proof one, by expansion (p. 246). Start from the defining limit. Expand the shifted power by the binomial theorem. Subtracting the unshifted power removes the leading term, and every surviving term carries at least one factor of the increment, so the increment can be taken outside as a common factor. Dividing by it leaves a sum whose first term is n times the (n − 1)th power of x and whose remaining terms all still carry the increment. Sending the increment to zero kills those and leaves the answer. Verified: the count is right — the binomial coefficient of the second term is exactly n, which is where the multiplier in the answer comes from.
• Proof two, by induction (p. 246). The case n = 1 is the derivative of x, already computed as the constant 1 on p. 245. For the step, write the nth power as x multiplied by the (n − 1)th power and apply the product rule: the first term gives the (n − 1)th power, the second gives x times (n − 1) times the (n − 2)th power by the inductive hypothesis, which is (n − 1) copies of the (n − 1)th power. Adding one copy to (n − 1) copies gives n of them. Verified: the arithmetic of the last line is 1 + (n − 1) = n, and that is the entire content of the step.
• The Remark under Theorem 6 (p. 246). The rule holds for any real exponent, and the chapter says outright that it will not prove that here. Exercise 12.2 q9 (iii), (iv) and (v), all with negative exponents, use the extension.
• Theorem 7 (§12.5.2, p. 246). For a polynomial with a non-zero leading coefficient and exponents running down from n, the derivative is the polynomial obtained by multiplying each term's coefficient by its exponent and reducing each exponent by one, the constant term dropping out. The chapter's proof is one sentence: it is the sum part of the derivative theorem on p. 244 together with Theorem 6. Note the printed typo: the exponent on the second term of the stated derivative is set as x with a superscript reading x − 2 where it should read n − 2. Confirmed on the p. 246 page image. State the rule correctly; do not reproduce the misprint.
• Example 13 (p. 246). The derivative of 6x¹⁰⁰ − x⁵⁵ + x. Verified: 600x⁹⁹ − 55x⁵⁴ + 1, each term by multiplying coefficient by exponent and dropping the exponent by one.
• Example 14 (p. 247). The derivative of 1 + x + x² + x³ + … + x⁵⁰, evaluated at x = 1. Verified: the derivative is 1 + 2x + 3x² + … + 50x⁴⁹, and at x = 1 every power is 1, so the value is the sum of the whole numbers from 1 to 50, which is 50 × 51 / 2 = 1275.
• Exercise 12.2 q5 (p. 248). The function whose terms are xᵏ divided by k for k running from 100 down to 2, followed by x and then 1; prove that its derivative at 1 is 100 times its derivative at 0. Verified: dividing xᵏ by k and then differentiating gives x^(k−1), so the derivative is x⁹⁹ + x⁹⁸ + … + x + 1, which is a hundred terms. At x = 1 each term is 1, giving 100. At x = 0 every term with a positive exponent vanishes and only the final 1 survives, giving 1. The stated relation follows.
• Exercise 12.2 q6 (p. 249). The derivative of xⁿ + a·xⁿ⁻¹ + a²·xⁿ⁻² + … + aⁿ⁻¹·x + aⁿ, for a fixed real a. This is Theorem 7 applied where the coefficients themselves are powers of a constant; the a factors are constants and ride through untouched.
• Exercise 12.2 q9 (p. 249), the power-rule items: (i) 2x − 3/4; (iii) x⁻³(5 + 3x); (iv) x⁵(3 − 6x⁻⁹); (v) x⁻⁴(3 − 4x⁻⁵). Each is expanded into a sum of powers first and then differentiated term by term — the expansion is the technique, and all three of (iii), (iv) and (v) need the Remark's extension to negative exponents.
• Exercise 12.2 q8 (p. 249). The derivative of (xⁿ − aⁿ)/(x − a) for a constant a. Note that this is the expression from Theorem 2 of §12.3.2 (p. 232); as a function of x it is a polynomial once the division is carried out, so the power rule applies to it directly.
• Miscellaneous Exercise items reachable this way (pp. 253–254): item 2 is a power of a shifted variable, and item 4 expands into a polynomial before differentiating. Item 25 does not belong on that list — it is a product with a cosine in one factor and a tangent in the other, so it needs the next topic's rules, and Rules for differentiating a sum, a product and a quotient files it correctly among the trigonometric items.

Figures to have open

• A binomial expansion laid out as a column of terms, with the leading term struck out and the shared factor of the increment highlighted down the rest. Standard schematic; §12.5.2 prints no figure, which I confirmed on pp. 245 to 248. Do not extend that to §12.5 as a whole — that section runs from p. 239 and carries Fig 12.11 on p. 241, the slope picture m03-t01 is built around. Counted across the chapter, captions fall on eleven pages only, running Fig 12.1 to Fig 12.11, and none of them is in §12.5.2.
• An induction ladder: n = 1 at the bottom, each rung labelled with the product rule and the previous rung.
• A polynomial-to-derivative transformation strip, each term morphing in place with its coefficient and exponent annotated.
• 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, Theorem 6 with its two proofs and the Remark, printed pp. 245–246; and §12.5.2 Derivative of polynomials and trigonometric functions, Theorem 7 with Examples 13 and 14, printed pp. 246–247.
• Exercise 12.2 items 5, 6, 8, 9, printed pp. 248–249.
• Miscellaneous Exercise on Chapter 12, items 2, 4, 25, printed pp. 253–254.
• Chapter Summary, p. 255, which lists the power rule among the standard derivatives.
• Backward reference inside the chapter: the expression differentiated in Exercise 12.2 q8 is the one whose limit is Theorem 2, §12.3.2, p. 232.