TMUA Differentiation — Practice Questions by Topic

These are SummitPapers original questions, not official past paper questions. Official TMUA questions sorted by topic are on TMUA past papers by topic.

MM6Differentiation
99 questions69 on Paper 1 · 30 on Paper 2
9 free solutionsThe rest show the correct letter only

Covers: stationary points and turning values, tangents and normals, increasing and decreasing functions, optimisation, limits used to find a gradient.

Paper 1 (69)Paper 2 (30)Practise this topic

Back to the syllabus · Differentiation · 1 of 99

Question 1Differentiation

Evaluate the limit, as x approaches 0, of (1 − cos(6x)) / x².

What this topic tests

Differentiation here means stationary points, tangents and normals, increasing and decreasing functions, optimisation, and a limit used to find a gradient.

How it is assessed

This is Section 1, on both papers in this set. Each question has five options. A paper is 20 questions in 75 minutes, with no calculator.

This note is written for SummitPapers. The official list of what can be examined is the specification, together with the Notes on Mathematics and, for Paper 2, the Notes on Logic and Proof.

Key methods

A stationary point

A stationary point has derivative zero. The first derivative changing from positive to negative, or the second derivative being negative, distinguishes a local maximum from a local minimum. A zero derivative alone does not say which.

Tangent and normal

The gradient of the tangent at a point is the derivative there. The normal is perpendicular to that tangent, so its gradient is the negative reciprocal when the tangent is not horizontal.

Common mistakes

The derivative of a product

The derivative of uv is u'v + uv', not u'v'. The derivative of u/v is (u'v − uv') / v².

A horizontal tangent treated as undefined

A zero derivative is a horizontal tangent. The normal is then vertical. It is not “no tangent”.

Worked example

This question is also in the list below, with the solution folded. It is opened here so the method is on the page before the other questions.

Given y = 4x³ − x, what is the value of dy/dx when x = 1?

A. 3
B. 4
C. 11
D. 12
E. 13

Answer: C. 11

Worked solution. Differentiate term by term: dy/dx = 12x² − 1. At x = 1 this is 12 − 1 = 11.

Why the other options look right. A substitutes x = 1 into 4x − 1 and never differentiates the cube. B reads off the coefficient 4 of x³ instead of differentiating and substituting x = 1. D is 12x² at x = 1, with the −1 omitted. E differentiates −x as +1, giving 12 + 1.

Paper 1 questions

Paper 1 is Applications of Mathematical Knowledge. Calculators are not allowed.

Q1. Evaluate the limit, as x approaches 0, of (1 − cos(6x)) / x².
Free · worked solution included
Answer and worked solution
Answer: D. 18
Worked solution. Use the identity 1 − cos(6x) = 2 sin²(3x). Then (1 − cos(6x))/x² = 2 (sin(3x)/x)² = 18 (sin(3x)/(3x))². As x approaches 0, sin(3x)/(3x) approaches 1, so the limit is 18.
Why the other options look right. A uses 1 − cos u ≈ u²/2 with u = 6x but writes (6x)² as 6x², so the limit is 6/2 = 3. B writes 1 − cos(6x) = 2 sin²(3x) but forgets to square sin(3x)/x, so it gets 2 × 3 = 6. C drops the factor 2 in the identity 1 − cos(6x) = 2 sin²(3x) and keeps only the limit of sin²(3x)/x², which is 9. E uses 1 − cos u ~ u² with u = 6x, giving 36 rather than half of that.
Q2. The curve y = x³ − 6x² − 288x + 4 has two stationary points. What is the greater of their x-coordinates?
Free · worked solution included
Answer and worked solution
Answer: E. 12
Worked solution. Differentiate: dy/dx = 3x² − 12x − 288 = 3(x² − 4x − 96) = 3(x − 12)(x + 8). The stationary points are at x = 12 and x = −8, and the greater x-coordinate is 12.
Why the other options look right. A applies the quadratic formula to x² − 4x − 96 = 0 but does not divide by 2a, giving 4 + 20 = 24. B reports the smaller stationary x-coordinate, −8, instead of the greater one. C reports the product of the roots of x² − 4x − 96 = 0, which is −96, instead of the greater root. D reports the sum of the roots of x² − 4x − 96 = 0, −b/a = 4, instead of the greater root.
Q3. A particle moves on a straight line. Its displacement in metres at time t seconds, for t ≥ 0, is s(t) = t³ − 3t² − 24t. Find all the times at which the particle is instantaneously at rest.
Free · worked solution included
Answer and worked solution
Answer: A. t = 4 only
Worked solution. The particle is at rest when the velocity is zero. Differentiate to get v(t) = 3t² − 6t − 24 = 3(t² − 2t − 8) = 3(t − 4)(t + 2). So v(t) = 0 at t = 4 and at t = −2. The motion is only defined for t ≥ 0, so t = −2 is rejected. The particle is instantaneously at rest at t = 4 only.
Why the other options look right. B solves v(t) = 0 correctly but keeps t = −2, which is outside t ≥ 0. C factors t² − 2t − 8 as (t − 2)(t + 4), with the signs swapped, which gives t = 2 or t = −4, and then keeps only t = 2. D sets the acceleration 6t − 6 equal to zero instead of the velocity, giving t = 1. E sets the displacement equal to zero: t(t² − 3t − 24) = 0 gives t = 0 or t = (3 ± √105)/2, and keeping the non-negative values gives t = 0 and t = (3 + √105)/2.
Q4. For x > 0, what is the maximum value of f(x) = x² e^−x?
Free · worked solution included
Answer and worked solution
Answer: A. 4/e²
Worked solution. The product rule gives f'(x) = 2x e^−x − x² e^−x = x(2 − x) e^−x. For x > 0 the only stationary point is x = 2. The factor x(2 − x) changes from positive to negative there, and f(x) tends to 0 at both ends of x > 0, so this stationary point is the maximum. Its value is f(2) = 4 e⁻² = 4/e².
Why the other options look right. B is the value of x e^−x at x = 2, which drops one factor of x. C replaces e⁻² by e⁻¹. D keeps only the factor e⁻². E evaluates x² e^−x at x = 1, the stationary point of x e^−x.
Q5. Find the second derivative of f(x) = (x + 1)e^3x.
Free · worked solution included
Answer and worked solution
Answer: A. (9x + 15)e^3x
Worked solution. The product rule gives f'(x) = e^3x + 3(x + 1)e^3x = (3x + 4)e^3x. Differentiating again, f''(x) = 3 e^3x + 3(3x + 4)e^3x = (3 + 9x + 12)e^3x = (9x + 15)e^3x.
Why the other options look right. B keeps only the chain-rule term 3(3x + 4)e^3x and drops the derivative of 3x + 4. C stops after one differentiation and reports f'(x) = (3x + 4)e^3x. D multiplies only one of the two product terms by 3, giving (3 + 3x + 4)e^3x. E keeps the correct bracket but replaces e^3x by e^x.
Q6. Find dy/dx if x² + y² = cos x.
Free · worked solution included
Answer and worked solution
Answer: E. −(2x + sin x)/(2y)
Worked solution. Differentiate both sides with respect to x. The left side gives 2x + 2y dy/dx, and the right side gives −sin x. So 2y dy/dx = −sin x − 2x, and dy/dx = −(2x + sin x)/(2y).
Why the other options look right. A reverses the overall sign. B differentiates cos x to sin x rather than to −sin x. C cancels the 2 in 2x against the 2 in the denominator 2y but leaves sin x unchanged, turning −(2x + sin x)/(2y) into −(x + sin x)/y. D omits the derivative of x².
Q7. The curve x² + xy + y² = 7 passes through (2, 1). What is dy/dx at that point?
Free · worked solution included
Answer and worked solution
Answer: C. −5/4
Worked solution. Differentiate both sides with respect to x. The derivative of x² is 2x, the derivative of xy is y + x dy/dx, and the derivative of y² is 2y dy/dx. So 2x + y + x dy/dx + 2y dy/dx = 0, and dy/dx = −(2x + y)/(x + 2y). At (2, 1) this is −(4 + 1)/(2 + 2) = −5/4. The point is on the curve because 4 + 2 + 1 = 7.
Why the other options look right. A differentiates xy as dy/dx alone, dropping both the factor x and the term y, so 2x + dy/dx + 2y dy/dx = 0 and dy/dx = −2x/(1 + 2y) = −4/3 at (2, 1). B differentiates xy as x dy/dx only, dropping the term y, so 2x + x dy/dx + 2y dy/dx = 0 and dy/dx = −2x/(x + 2y) = −4/4 = −1. D differentiates xy as y only, dropping the term x dy/dx, so 2x + y + 2y dy/dx = 0 and dy/dx = −(2x + y)/(2y) = −5/2. E moves 2x + y to the other side without changing its sign, giving dy/dx = (2x + y)/(x + 2y) = 5/4.
Q8. For x > 0, let y = x^3x. Which expression is equal to dy/dx?
Free · worked solution included
Answer and worked solution
Answer: A. 3 x^3x (1 + ln x)
Worked solution. Write y = e^{3x ln x}. Differentiating the exponent gives 3 ln x + 3x × (1/x) = 3 ln x + 3 = 3(1 + ln x). Therefore dy/dx = 3 x^3x (1 + ln x).
Why the other options look right. B drops the factor 3 that comes from the exponent 3x. C keeps that factor but drops the term 3x × (1/x) = 3 that comes from differentiating ln x in the product 3x ln x. D treats the exponent 3x as a constant and applies the ordinary power rule, giving 3x × x^{3x − 1} = 3 x^3x. E differentiates x ln x as ln x + 1/x, forgetting to multiply 1/x by x, giving 3 x^3x (ln x + 1/x).
Q9. For x > 0, which expression is the derivative of x³ − 4x² + 6x√x with respect to x?
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Answer
Answer: E. 3x² − 8x + 9√x
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Q10. A curve is given parametrically by x = t² − 4t and y = t³ − 3t², with t real. At how many points does this curve have a horizontal tangent?
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Answer
Answer: B. 1
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Q11. The polynomial f(x) = x³ + px² + qx + r satisfies f(1) = 10, f'(−1) = 7 and f''(0) = 2p. What is p + q + r?
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Answer
Answer: B. 9
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Q12. How many distinct real solutions does (x³ − 3x)² = 1 have?
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Answer
Answer: E. 6
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Q13. For a positive number a, let I(a) = ∫ from 0 to a of (12 − 3x²) dx. For which a is dI/da = 0?
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Answer
Answer: A. 2
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Q14. The normal to the curve y = 3/x² at the point where x = 1 meets the coordinate axes at P and Q. Find the distance PQ.
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Answer
Answer: E. 17√37 / 6
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Q15. An open-top box has a rectangular base whose length is twice its width. Its volume is 36 cm³. What is the minimum possible surface area of the box?
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Answer
Answer: A. 54 cm²
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Q16. The function f is given by f(x) = (x^3/2 − 2)² / x for x > 0. What is the value of f''(1)?
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Answer
Answer: D. 11
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Q17. The curve C has equation y = x³ − p x² + p², where p is real. The tangent to C at the point where x = 2 meets the y-axis at (0, c). What is the least possible value of c as p varies?
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Answer
Answer: D. −20
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Q18. A point P lies on the parabola y = x² + 1. What is the minimum possible distance from P to the point (0, 4)?
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Answer
Answer: A. √11/2
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Q19. The line y = mx + 2, where m > 0, is normal to the curve y = 9 − (1/2)x² at the point (p, q). What is p?
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Answer
Answer: D. 2√3
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Q20. The derivative f' is continuous, and f'(x) = 0 only at the five points described here. At x = −2 the graph of f' touches the x-axis, and f' is positive on both sides. At x = 1, f' changes from positive to negative. At x = 4 the graph of f' touches the x-axis, and f' is negative on both sides. At x = 6, f' changes from negative to positive. At x = 9, f' changes from positive to negative. How many local maxima does f have?
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Answer
Answer: B. 2
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Q21. A curve has equation y = (x − q)²(2x + 1). The gradient at x = 1 depends on q. Which value of q minimises that gradient?
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Answer
Answer: B. 5/2
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Q22. For x > 0, what is the derivative of (x³ − 7x²)/(2x√x)?
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Answer
Answer: C. (3x − 7)/(4√x)
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Q23. Let f(x) = √x (4x² − 3x + 1). For what fraction of the interval 0 < x < 1 is f decreasing?
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Answer
Answer: A. 1/20
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Q24. The curve y = x³ − 3x has a tangent of gradient 9 at the point where x = a, with a > 0. What is the y-intercept of this tangent?
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Answer
Answer: B. −16
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Q25. The function f satisfies f''(x) = a for every x, f(0) = 4 and f(1) = 0, and the integral of f from 0 to 1 equals 1. Find a.
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Answer
Answer: E. 12
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Q26. For each real m, let y = 2x² − 2mx + (3/2)m² + 6m + 4. Among all points on all of these curves, the one with the smallest y-coordinate is (a, b). Find a + b.
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Answer
Answer: E. −13/2
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Q27. A rectangle with sides parallel to the axes sits in the region between y = 6 − x² and y = x² − 2. What is the greatest possible area of the rectangle?
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Answer
Answer: C. 64√3/9
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Q28. The equation x³ − 12x + b = 0 has three distinct real roots, exactly two of which are positive. What is the complete range of possible values of b?
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Answer
Answer: E. 0 < b < 16
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Q29. Let f(x) = x² − 4x + 7. What are the coordinates of the turning point of y = 1 − 2f(x − 2)?
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Answer
Answer: A. (4, −5)
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Q30. Given y = 4x³ − x, what is the value of dy/dx when x = 1?
Free · worked solution included
Answer and worked solution
Answer: C. 11
Worked solution. Differentiate term by term: dy/dx = 12x² − 1. At x = 1 this is 12 − 1 = 11.
Why the other options look right. A substitutes x = 1 into 4x − 1 and never differentiates the cube. B reads off the coefficient 4 of x³ instead of differentiating and substituting x = 1. D is 12x² at x = 1, with the −1 omitted. E differentiates −x as +1, giving 12 + 1.
Q31. Starting from f₁(x) = 3x, each following function is 3x times the derivative of the function before it. Find f₁(x) + f₂(x) + … + f₁₀(x).
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Answer
Answer: B. (3x/2)(3¹⁰ − 1)
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Q32. For x > 0, the tangent to y = ln x at x = a passes through the origin. Find a.
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Answer
Answer: D. e
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Q33. How many distinct real roots does the equation x⁵ − 5x³ − 20x + 60 = 0 have?
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Answer
Answer: A. 1
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Q34. Find the x-coordinate of the local maximum of the curve y = 2x³ − 3x² − 12x + 5.
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Answer
Answer: E. −1
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Q35. The curve y = x³ + 3x² − 189x + 1 has a local maximum at which value of x?
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Answer
Answer: A. x = −9
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Q36. The curve y = 2x³ + 3x² − 36x + 5 has a local maximum. At what value of x does it occur?
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Answer
Answer: A. x = −3
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Q37. Let f(x) = x² + 6x + 10. Find the turning point of y = 2f(x − 1) + 3.
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Answer
Answer: A. (−2, 5)
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Q38. The curve y = x³ + 6x² − 231x + 5 has a local maximum at x equal to
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Answer
Answer: C. −11
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Q39. At which value of x does y = x³ − 6x² − 135x + 2 have a local maximum?
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Answer
Answer: B. x = −5
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Q40. At which value of x does the curve y = x³ − 12x² − 60x + 1 have a local maximum?
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Answer
Answer: B. x = −2
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Q41. The curve y = x³ − 3x² − 45x + 4 has a local maximum at which value of x?
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Answer
Answer: D. x = −3
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Q42. Let g(x) = (3x − 1)√x for x > 0. What is the value of g'(4)?
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Answer
Answer: D. 35/4
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Q43. The curve y = x⁴ − x² has a straight line that is tangent to the curve at two distinct points. What is the equation of that line?
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Answer
Answer: B. y = −1/4
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Q44. The function f is given by f(x) = (2/x − 3/x²)², where x ≠ 0. Find f''(1).
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Answer
Answer: D. 60
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Q45. The curve C has equation y = 2x² + kx + 4, where k ≥ 0. Find the value of k which minimises the distance from the origin to the stationary point of C.
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Answer
Answer: C. k = √30
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Q46. Let f(x) = x² − 5x + 7. What are the coordinates of the turning point of y = 4 − 3 f(x − 2)?
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Answer
Answer: B. (9/2, 7/4)
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Q47. Define f₁(x) = x² and f_n+1(x) = x f_n'(x) for n ≥ 1. What is f₁(x) + f₂(x) + ... + f₈(x)?
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Answer
Answer: A. 255 x²
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Q48. The curve x² + 2xy + 3y² = 17 passes through (1, 2). What is the gradient of the curve at this point?
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Answer
Answer: A. −3/7
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Q49. How many distinct real roots does x⁵ − 5x + 1 = 0 have?
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Answer
Answer: C. 3
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Q50. The curve y = (x² + 3√x)(x − 2) / √x is defined for x > 0. Find its gradient at x = 4.
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Answer
Answer: C. 17
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Q51. A curve has equation y = (3x² + 4)/(x√x), for x > 0. Find the gradient at x = 4.
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Answer
Answer: E. 9/16
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Q52. The curve y = x³ − 3px² + 3(p + 2)x + 1 has two distinct turning points, and both of them have positive x-coordinates. Find the complete set of possible values of p.
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Answer
Answer: A. p > 2
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Q53. Find the complete set of values of k for which the curve y = (x² + k)/(x − 1) has two distinct stationary points.
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Answer
Answer: D. k > −1
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Q54. The cubic f(x) = x³ + ax² + bx + 7 has exactly one stationary point, and that stationary point is at x = 2. Find a + b.
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Answer
Answer: B. 6
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Q55. Let f(x) = x^1/3(x − 3)². The fraction of the interval 0 < x < 6 on which f is decreasing is
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Answer
Answer: A. 3/7
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Q56. A curve has equation y = 2x² + 3 and a line has equation y = 5x − 1. What is the shortest distance, measured parallel to the y-axis, between the curve and the line?
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Answer
Answer: D. 7/8
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Q57. A curve has equation y = x³ − 4x² + a, where a is a constant. The tangent at x = 3 and the normal at x = 1 meet on the x-axis. Find a.
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Answer
Answer: B. 15/7
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Q58. The point P lies on the curve y = x² and is as close as possible to Q(−6, 3). Find the distance PQ.
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Answer
Answer: E. √17
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Q59. How many distinct real solutions does the equation x⁴ − 4x³ − 2x² + 12x + 5 = 0 have?
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Answer
Answer: C. 4
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Q60. What is the term of highest degree in d²/dx²[(x² + 3)²(x² − 1)²] − d/dx[(x² + 1)²(3x³ − 1)]?
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Answer
Answer: B. 35x⁶
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Q61. P is a point on the curve y = 3 − x² with positive x-coordinate. The tangent to the curve at P meets the x-axis at M and the y-axis at N, and O is the origin. Find the least possible area of triangle OMN.
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Answer
Answer: C. 4
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Q62. A curve has equation y = 2x³ + 3px² + (p² + 4)x, where p is real. Let M be the gradient of the normal at x = 1. What is the least possible value of M?
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Answer
Answer: A. −1
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Q63. How many distinct real solutions does the equation 3x⁵ − 5x³ + 1 = 0 have?
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Answer
Answer: C. 3
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Q64. The curve y = x³ + bx² + cx + 1, where b and c are real, has a local maximum at x = 1. Which statement must be true?
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Answer
Answer: E. b < −3
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Q65. Let f(x) = (x − 1)²/(x²)^1/3 for every real x ≠ 0. On which complete set of values of x is f decreasing?
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Answer: E. x < −1/2 or 0 < x < 1
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Q66. The volume of a balloon is modelled by V = (a − bt)³, where a and b are positive constants and t is the time in seconds, with t ≥ 0. When t = 2 the volume is 27 cm³ and it is decreasing at 18 cm³ per second. Find a.
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Answer
Answer: C. 13/3
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Q67. The least possible value of the gradient of y = (x + a)²(2x + a), at the point where x = 1, as a varies, is
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Answer
Answer: A. −1/4
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Q68. A differentiable function f satisfies f'(x) = (x − 4)²(x + 3) for every real x. Which statement is true?
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Answer
Answer: B. f has a local minimum at x = −3.
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Q69. Let f(x) = 3x⁴ − 8x³ − 6x² + 24x. Find the complete set of values of k for which the equation f(x) = k has four distinct real solutions.
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Answer: B. 8 < k < 13
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Paper 2 questions

Paper 2 is Mathematical Reasoning. It can test this same topic. Argument, proof, and identifying errors are the topics that appear on Paper 2 only.

Q1. The tangent to y = x² at (1, 1) is parallel to the line L, and the distance between the tangent and L is √5. What is the equation of L?
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Answer
Answer: C. 2x − y + 4 = 0 or 2x − y − 6 = 0
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Q2. A coincidence point of a function f is a real number x such that f(x) = f'(x). Let f(x) = x⁴ + k, where k is a real constant. Which describes all the values of k for which f has exactly two coincidence points?
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Answer: E. k < 27
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Q3. For which values of the constant k do exactly three different tangents to the curve y = x³ − 6x pass through the point (1, k)?
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Answer: A. −6 < k < −5
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Q4. The function f is differentiable for all real x, f(1) = 0, and f'(x) > f(x) for every real x. What is the complete set of x for which (x² − 4) f(x) < 0?
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Answer: D. (−∞, −2) ∪ (1, 2)
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Q5. For x > 0, let f(x) = 2(x⁵ + 3x) / (x⁵)^1/4. Find f'(x).
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Answer: A. (15/2)x^11/4 − (3/2)x^−5/4
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Q6. Given that y = (1 − 2x)² / (2 x^3/2), which one of the following is a correct expression for dy/dx?
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Answer: A. x^−1/2 + x^−3/2 − (3/4)x^−5/2
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Q7. The side length of a cube is increasing at 2 cm per second. At the instant when the side length is 3 cm, at what rate is the volume increasing?
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Answer: D. 54 cm³/s
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Q8. The function f is given for x > 0 by f(x) = (x³ − 9x) / (3√x). Find f'(9).
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Answer: C. 22
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Q9. The function f is a polynomial defined for every real x. Which of the following is necessary and sufficient for f(b) − f(a) ≥ b − a to hold for all real a and b with a < b?
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Answer: C. f'(x) ≥ 1 for every real x
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Q10. The polynomial p(x) is increasing for a ≤ x ≤ b, where a < b. Which of the following must be true? I. p'(x) ≥ 0 for a ≤ x ≤ b II. p(x) ≥ p(a) for a ≤ x ≤ b III. (p(x))² is increasing for a ≤ x ≤ b
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Answer: D. I and II only
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Q11. How many stationary points does y = 5x⁴ + 2x³ + 3x² − 1 have?
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Answer: B. 1
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Q12. Let f be a non-constant polynomial. Suppose f(x) = 0 for exactly M distinct real x, and f'(x) = 0 for exactly N distinct real x. Which of these statements are true? I: It is possible that M = N + 2. II: It is possible that M = N + 1. III: It is possible that N = M + 3.
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Answer: E. II and III only
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Q13. What is the gradient of y = (2x − 3)² / (x √x) at x = 9?
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Answer: B. 5/6
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Q14. The polynomial y = a x⁴ + b x³ + c x² + d x + e has real coefficients and a ≠ 0. Which of the following is not possible?
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Answer: D. The graph has one local minimum and one local maximum.
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Q15. The parabola y = x² + 4ax + 3 has one turning point. For which values of a is that turning point as close as possible to the origin?
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Answer: D. ±√(5/8)
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Q16. The curve y = (3 − x)(1 + x) and the curve y = (a − x)(a + x) each have a maximum point. Find the values of the constant a for which these two maximum points are as close as possible.
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Answer: C. ±2
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Q17. For x > 0, the function f(x) = x + 36/x has a minimum value. What is this minimum value?
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Answer: B. 12
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Q18. For x > 0, let f(x) = x(3 − ln x). What is the greatest value of f(x)?
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Answer: A. e²
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Q19. How many real solutions does the equation x³ + 3x − 2 = 0 have?
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Answer: B. 1
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Q20. The function f(x) = x⁴ − 8x² + 3 has two local minima. What is the value of f at either local minimum?
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Answer: A. −13
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Q21. The curve y = x² + 3 and the line y = 2x do not meet. What is the shortest distance between a point of the curve and a point of the line?
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Answer: B. (2√5)/5
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Q22. For 0 ≤ x ≤ 9, let f(x) = x√(9 − x). What is the greatest value of f(x)?
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Answer: D. 6√3
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Q23. For x ≥ 0, let f(x) = x³(25 − x²). At the maximum point of f, what is the value of x²?
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Answer: C. 15
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Q24. An open box is made by cutting squares of side x cm from each corner of a 12 cm by 12 cm sheet and folding up the sides. What is the greatest possible volume of the box?
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Answer: C. 128 cm³
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Q25. The function f is differentiable for every real x. Let P be the statement “f'(2) = 0”, and let Q be the statement “f has a turning point at x = 2”. Which of the following is correct?
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Answer: B. P is necessary but not sufficient for Q
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Q26. Find the minimum value of f(x) = x³ − 3x² − 9x + 10 for 0 ≤ x ≤ 4.
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Answer
Answer: A. −17
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Q27. Find the values of a for which the turning point of y = x² + 2 a x + 4 is as close as possible to the origin.
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Answer: E. a = ± √(7/2)
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Q28. The curve y = 5 − (x − 2a)² and the curve y = a − x² each have a maximum point. Find the value of a for which these two maximum points are as close as possible.
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Answer: C. 1
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Q29. The function f is defined for all real x by f(x) = (x + 1)/(x² + 3). What is the difference between the greatest value and the least value of f(x)?
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Answer: A. 2/3
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Q30. For how many of these functions is the derivative strictly increasing for every real x? The functions are e^x, sin x, x², x³ and cos x.
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Answer: C. 2
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Evaluate the limit, as x approaches 0, of (1 − cos(6x)) / x2.

What this topic tests

How it is assessed

Key methods

A stationary point

Tangent and normal

Common mistakes

The derivative of a product

A horizontal tangent treated as undefined

Worked example

Given y = 4x³ − x, what is the value of dy/dx when x = 1?

Paper 1 questions

Q1. Evaluate the limit, as x approaches 0, of (1 − cos(6x)) / x2.

Q2. The curve y = x3 − 6x2 − 288x + 4 has two stationary points. What is the greater of their x-coordinates?

Q3. A particle moves on a straight line. Its displacement in metres at time t seconds, for t ≥ 0, is s(t) = t3 − 3t2 − 24t. Find all the times at which the particle is instantaneously at rest.

Q4. For x > 0, what is the maximum value of f(x) = x2 e−x?

Q5. Find the second derivative of f(x) = (x + 1)e3x.

Q6. Find dy/dx if x2 + y2 = cos x.

Q7. The curve x2 + xy + y2 = 7 passes through (2, 1). What is dy/dx at that point?

Q8. For x > 0, let y = x3x. Which expression is equal to dy/dx?

Q9. For x > 0, which expression is the derivative of x3 − 4x2 + 6x√x with respect to x?

Q10. A curve is given parametrically by x = t2 − 4t and y = t3 − 3t2, with t real. At how many points does this curve have a horizontal tangent?

Q11. The polynomial f(x) = x3 + px2 + qx + r satisfies f(1) = 10, f'(−1) = 7 and f''(0) = 2p. What is p + q + r?

Q12. How many distinct real solutions does (x3 − 3x)2 = 1 have?

Q13. For a positive number a, let I(a) = ∫ from 0 to a of (12 − 3x2) dx. For which a is dI/da = 0?

Q14. The normal to the curve y = 3/x2 at the point where x = 1 meets the coordinate axes at P and Q. Find the distance PQ.

Q15. An open-top box has a rectangular base whose length is twice its width. Its volume is 36 cm3. What is the minimum possible surface area of the box?

Q16. The function f is given by f(x) = (x3/2 − 2)2 / x for x > 0. What is the value of f''(1)?

Q17. The curve C has equation y = x3 − p x2 + p2, where p is real. The tangent to C at the point where x = 2 meets the y-axis at (0, c). What is the least possible value of c as p varies?

Q18. A point P lies on the parabola y = x² + 1. What is the minimum possible distance from P to the point (0, 4)?

Q19. The line y = mx + 2, where m > 0, is normal to the curve y = 9 − (1/2)x2 at the point (p, q). What is p?

Q21. A curve has equation y = (x − q)²(2x + 1). The gradient at x = 1 depends on q. Which value of q minimises that gradient?

Q22. For x > 0, what is the derivative of (x3 − 7x2)/(2x√x)?

Q23. Let f(x) = √x (4x2 − 3x + 1). For what fraction of the interval 0 < x < 1 is f decreasing?

Q24. The curve y = x³ − 3x has a tangent of gradient 9 at the point where x = a, with a > 0. What is the y-intercept of this tangent?

Q25. The function f satisfies f''(x) = a for every x, f(0) = 4 and f(1) = 0, and the integral of f from 0 to 1 equals 1. Find a.

Q26. For each real m, let y = 2x2 − 2mx + (3/2)m2 + 6m + 4. Among all points on all of these curves, the one with the smallest y-coordinate is (a, b). Find a + b.

Q27. A rectangle with sides parallel to the axes sits in the region between y = 6 − x2 and y = x2 − 2. What is the greatest possible area of the rectangle?

Q28. The equation x³ − 12x + b = 0 has three distinct real roots, exactly two of which are positive. What is the complete range of possible values of b?

Q29. Let f(x) = x² − 4x + 7. What are the coordinates of the turning point of y = 1 − 2f(x − 2)?

Q30. Given y = 4x³ − x, what is the value of dy/dx when x = 1?

Q31. Starting from f1(x) = 3x, each following function is 3x times the derivative of the function before it. Find f1(x) + f2(x) + … + f10(x).

Q32. For x > 0, the tangent to y = ln x at x = a passes through the origin. Find a.

Q33. How many distinct real roots does the equation x5 − 5x3 − 20x + 60 = 0 have?

Q34. Find the x-coordinate of the local maximum of the curve y = 2x³ − 3x² − 12x + 5.

Q35. The curve y = x³ + 3x² − 189x + 1 has a local maximum at which value of x?

Q36. The curve y = 2x³ + 3x² − 36x + 5 has a local maximum. At what value of x does it occur?

Q37. Let f(x) = x² + 6x + 10. Find the turning point of y = 2f(x − 1) + 3.

Q38. The curve y = x³ + 6x² − 231x + 5 has a local maximum at x equal to

Q39. At which value of x does y = x³ − 6x² − 135x + 2 have a local maximum?

Q40. At which value of x does the curve y = x³ − 12x² − 60x + 1 have a local maximum?

Q41. The curve y = x³ − 3x² − 45x + 4 has a local maximum at which value of x?

Q42. Let g(x) = (3x − 1)√x for x > 0. What is the value of g'(4)?

Q43. The curve y = x4 − x2 has a straight line that is tangent to the curve at two distinct points. What is the equation of that line?

Q44. The function f is given by f(x) = (2/x − 3/x2)2, where x ≠ 0. Find f''(1).

Q45. The curve C has equation y = 2x2 + kx + 4, where k ≥ 0. Find the value of k which minimises the distance from the origin to the stationary point of C.

Q46. Let f(x) = x2 − 5x + 7. What are the coordinates of the turning point of y = 4 − 3 f(x − 2)?

Q47. Define f1(x) = x2 and fn+1(x) = x fn'(x) for n ≥ 1. What is f1(x) + f2(x) + ... + f8(x)?

Q48. The curve x² + 2xy + 3y² = 17 passes through (1, 2). What is the gradient of the curve at this point?

Q49. How many distinct real roots does x5 − 5x + 1 = 0 have?

Q50. The curve y = (x² + 3√x)(x − 2) / √x is defined for x > 0. Find its gradient at x = 4.

Q51. A curve has equation y = (3x² + 4)/(x√x), for x > 0. Find the gradient at x = 4.

Q52. The curve y = x³ − 3px² + 3(p + 2)x + 1 has two distinct turning points, and both of them have positive x-coordinates. Find the complete set of possible values of p.

Q53. Find the complete set of values of k for which the curve y = (x² + k)/(x − 1) has two distinct stationary points.

Q54. The cubic f(x) = x³ + ax² + bx + 7 has exactly one stationary point, and that stationary point is at x = 2. Find a + b.

Q55. Let f(x) = x1/3(x − 3)². The fraction of the interval 0 < x < 6 on which f is decreasing is

Q56. A curve has equation y = 2x² + 3 and a line has equation y = 5x − 1. What is the shortest distance, measured parallel to the y-axis, between the curve and the line?

Q57. A curve has equation y = x³ − 4x² + a, where a is a constant. The tangent at x = 3 and the normal at x = 1 meet on the x-axis. Find a.

Q58. The point P lies on the curve y = x² and is as close as possible to Q(−6, 3). Find the distance PQ.

Q59. How many distinct real solutions does the equation x⁴ − 4x³ − 2x² + 12x + 5 = 0 have?

Q60. What is the term of highest degree in d²/dx²[(x² + 3)²(x² − 1)²] − d/dx[(x² + 1)²(3x³ − 1)]?

Q61. P is a point on the curve y = 3 − x² with positive x-coordinate. The tangent to the curve at P meets the x-axis at M and the y-axis at N, and O is the origin. Find the least possible area of triangle OMN.

Q62. A curve has equation y = 2x³ + 3px² + (p² + 4)x, where p is real. Let M be the gradient of the normal at x = 1. What is the least possible value of M?

Q63. How many distinct real solutions does the equation 3x⁵ − 5x³ + 1 = 0 have?

Q64. The curve y = x³ + bx² + cx + 1, where b and c are real, has a local maximum at x = 1. Which statement must be true?

Q65. Let f(x) = (x − 1)²/(x²)1/3 for every real x ≠ 0. On which complete set of values of x is f decreasing?

Q66. The volume of a balloon is modelled by V = (a − bt)³, where a and b are positive constants and t is the time in seconds, with t ≥ 0. When t = 2 the volume is 27 cm³ and it is decreasing at 18 cm³ per second. Find a.

Q67. The least possible value of the gradient of y = (x + a)²(2x + a), at the point where x = 1, as a varies, is

Q68. A differentiable function f satisfies f'(x) = (x − 4)²(x + 3) for every real x. Which statement is true?

Q69. Let f(x) = 3x⁴ − 8x³ − 6x² + 24x. Find the complete set of values of k for which the equation f(x) = k has four distinct real solutions.

Evaluate the limit, as x approaches 0, of (1 − cos(6x)) / x².

Q1. Evaluate the limit, as x approaches 0, of (1 − cos(6x)) / x².

Q2. The curve y = x³ − 6x² − 288x + 4 has two stationary points. What is the greater of their x-coordinates?

Q3. A particle moves on a straight line. Its displacement in metres at time t seconds, for t ≥ 0, is s(t) = t³ − 3t² − 24t. Find all the times at which the particle is instantaneously at rest.

Q4. For x > 0, what is the maximum value of f(x) = x² e^−x?

Q5. Find the second derivative of f(x) = (x + 1)e^3x.

Q6. Find dy/dx if x² + y² = cos x.

Q7. The curve x² + xy + y² = 7 passes through (2, 1). What is dy/dx at that point?

Q8. For x > 0, let y = x^3x. Which expression is equal to dy/dx?

Q9. For x > 0, which expression is the derivative of x³ − 4x² + 6x√x with respect to x?

Q10. A curve is given parametrically by x = t² − 4t and y = t³ − 3t², with t real. At how many points does this curve have a horizontal tangent?

Q11. The polynomial f(x) = x³ + px² + qx + r satisfies f(1) = 10, f'(−1) = 7 and f''(0) = 2p. What is p + q + r?

Q12. How many distinct real solutions does (x³ − 3x)² = 1 have?

Q13. For a positive number a, let I(a) = ∫ from 0 to a of (12 − 3x²) dx. For which a is dI/da = 0?

Q14. The normal to the curve y = 3/x² at the point where x = 1 meets the coordinate axes at P and Q. Find the distance PQ.

Q15. An open-top box has a rectangular base whose length is twice its width. Its volume is 36 cm³. What is the minimum possible surface area of the box?

Q16. The function f is given by f(x) = (x^3/2 − 2)² / x for x > 0. What is the value of f''(1)?

Q17. The curve C has equation y = x³ − p x² + p², where p is real. The tangent to C at the point where x = 2 meets the y-axis at (0, c). What is the least possible value of c as p varies?

Q19. The line y = mx + 2, where m > 0, is normal to the curve y = 9 − (1/2)x² at the point (p, q). What is p?

Q22. For x > 0, what is the derivative of (x³ − 7x²)/(2x√x)?

Q23. Let f(x) = √x (4x² − 3x + 1). For what fraction of the interval 0 < x < 1 is f decreasing?

Q26. For each real m, let y = 2x² − 2mx + (3/2)m² + 6m + 4. Among all points on all of these curves, the one with the smallest y-coordinate is (a, b). Find a + b.

Q27. A rectangle with sides parallel to the axes sits in the region between y = 6 − x² and y = x² − 2. What is the greatest possible area of the rectangle?

Q31. Starting from f₁(x) = 3x, each following function is 3x times the derivative of the function before it. Find f₁(x) + f₂(x) + … + f₁₀(x).

Q33. How many distinct real roots does the equation x⁵ − 5x³ − 20x + 60 = 0 have?

Q43. The curve y = x⁴ − x² has a straight line that is tangent to the curve at two distinct points. What is the equation of that line?

Q44. The function f is given by f(x) = (2/x − 3/x²)², where x ≠ 0. Find f''(1).

Q45. The curve C has equation y = 2x² + kx + 4, where k ≥ 0. Find the value of k which minimises the distance from the origin to the stationary point of C.

Q46. Let f(x) = x² − 5x + 7. What are the coordinates of the turning point of y = 4 − 3 f(x − 2)?

Q47. Define f₁(x) = x² and f_n+1(x) = x f_n'(x) for n ≥ 1. What is f₁(x) + f₂(x) + ... + f₈(x)?

Q49. How many distinct real roots does x⁵ − 5x + 1 = 0 have?

Q55. Let f(x) = x^1/3(x − 3)². The fraction of the interval 0 < x < 6 on which f is decreasing is

Q65. Let f(x) = (x − 1)²/(x²)^1/3 for every real x ≠ 0. On which complete set of values of x is f decreasing?

Q1. The tangent to y = x² at (1, 1) is parallel to the line L, and the distance between the tangent and L is √5. What is the equation of L?

Q2. A coincidence point of a function f is a real number x such that f(x) = f'(x). Let f(x) = x⁴ + k, where k is a real constant. Which describes all the values of k for which f has exactly two coincidence points?

Q3. For which values of the constant k do exactly three different tangents to the curve y = x³ − 6x pass through the point (1, k)?

Q4. The function f is differentiable for all real x, f(1) = 0, and f'(x) > f(x) for every real x. What is the complete set of x for which (x² − 4) f(x) < 0?

Q5. For x > 0, let f(x) = 2(x⁵ + 3x) / (x⁵)^1/4. Find f'(x).

Q6. Given that y = (1 − 2x)² / (2 x^3/2), which one of the following is a correct expression for dy/dx?

Q8. The function f is given for x > 0 by f(x) = (x³ − 9x) / (3√x). Find f'(9).

Q11. How many stationary points does y = 5x⁴ + 2x³ + 3x² − 1 have?

Q13. What is the gradient of y = (2x − 3)² / (x √x) at x = 9?

Q14. The polynomial y = a x⁴ + b x³ + c x² + d x + e has real coefficients and a ≠ 0. Which of the following is not possible?

Q15. The parabola y = x² + 4ax + 3 has one turning point. For which values of a is that turning point as close as possible to the origin?

Q26. Find the minimum value of f(x) = x³ − 3x² − 9x + 10 for 0 ≤ x ≤ 4.

Q27. Find the values of a for which the turning point of y = x² + 2 a x + 4 is as close as possible to the origin.

Q28. The curve y = 5 − (x − 2a)² and the curve y = a − x² each have a maximum point. Find the value of a for which these two maximum points are as close as possible.

Q30. For how many of these functions is the derivative strictly increasing for every real x? The functions are e^x, sin x, x², x³ and cos x.