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The product, quotient and chain rules questions
Year 12 differentiated by rewriting; that only stretches so far. Three rules finish the job properly: the chain rule for functions inside functions, the product rule for multiplied pairs, and the quotient rule for fractions, and between them they differentiate everything this course can write down.
9 original questions · 30 marks · the the product, quotient and chain rules notes · Differentiation
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening the worked one: the answers award marks point by point, and the marks are easier to see when you have something of your own to compare against.
Differentiate y = (2x + 1)4.
Worked answer
Differentiate the outside first, leaving the inside untouched, to get 4(2x + 1)3, then multiply by the inside's derivative, 2. So dy/dx = 8(2x + 1)3. M1 A1. Rates through a chain multiply, and the missing factor of 2 is the single most common omission in the topic.Differentiate y = ex².
Worked answer
The outside is e to something, which is its own derivative, and the inside x2 contributes 2x. So dy/dx = 2x ex². M1 A1. The exponential never changes shape, so the chain factor is doing all the varying.Differentiate y = (x2 − 3)6, and find the coordinates of the stationary points of the curve.
Worked answer
With u = x2 − 3, dy/du = 6u5 and du/dx = 2x, so dy/dx = 12x(x2 − 3)5. Setting this to zero gives x = 0 or x2 = 3, so x = 0, √3 and −√3. The corresponding y values are (−3)6 = 729 at x = 0 and 0 at both x = ±√3, so the stationary points are (0, 729), (√3, 0) and (−√3, 0). M1 for the chain rule, A1 for the derivative, M1 for solving, A1 for all three x values, A1 for the points. Setting only the bracket to zero misses x = 0 entirely, and a product is zero when either factor is.Differentiate y = x3e2x.
Worked answer
Product rule with u = x3 and v = e2x: dy/dx = 3x2e2x + 2x3e2x, which factorises as x2e2x(3 + 2x). M1 for the rule, A1 for the two terms, A1 for the factorised form. Each factor takes a turn being differentiated while the other one waits, and the chain rule is still needed inside the second term to produce the 2.Differentiate y = x sin x, and find the gradient of the curve at x = π.
Worked answer
Product rule: dy/dx = 1 × sin x + x × cos x = sin x + x cos x. At x = π, sin π = 0 and cos π = −1, so the gradient is 0 + π(−1) = −π. M1 for the rule, A1 for the derivative, A1 for −π. Differentiating each factor and multiplying the results would give cos x on its own, and that is wrong. The rule shares the differentiation out one factor at a time.Differentiate y = (2x + 1)/(x2 + 1), and find the gradient of the curve at x = 0.
Worked answer
Quotient rule with u = 2x + 1 and v = x2 + 1: dy/dx = [2(x2 + 1) − (2x + 1)(2x)]/(x2 + 1)2, which simplifies to (−2x2 − 2x + 2)/(x2 + 1)2. At x = 0 the gradient is 2/1 = 2. M1 for the rule, A1 for the numerator, A1 for the simplification, A1 for the value. The minus sign makes the order u′v − uv′ unforgiving, and swapping the two products flips the sign of every answer you will ever get from it.Differentiate y = sin3 x.
Worked answer
Read the expression as (sin x)3. The outside gives 3(sin x)2 and the inside gives cos x, so dy/dx = 3 sin2 x cos x. M1 A1. The notation sin3 x hides the chain, and rewriting it with a bracket makes both layers visible before you start.Given that y = x2 ln 3x for x > 0, show that dy/dx = x(2 ln 3x + 1), and find the exact coordinates of the stationary point of the curve.
Worked answer
Product rule with u = x2 and v = ln 3x. Note that ln 3x = ln 3 + ln x, so its derivative is 1/x. Then dy/dx = 2x ln 3x + x2 × 1/x = 2x ln 3x + x = x(2 ln 3x + 1), as required. At a stationary point, x > 0 so the bracket must vanish: 2 ln 3x + 1 = 0 gives ln 3x = −½, so 3x = e−½ and x = 1/(3√e). Then y = x2 ln 3x = (1/(9e)) × (−½) = −1/(18e). The stationary point is (1/(3√e), −1/(18e)). M1 for the product rule, A1 for each term, A1 for the printed form, M1 for solving the bracket, A1 for x, A1 for y. The derivative of ln 3x is 1/x rather than 1/(3x). The chain rule gives 3/(3x), and the threes cancel.By writing tan x as sin x/cos x, use the quotient rule to show that the derivative of tan x is sec2 x.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
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