Lesson G1

G1 Derivative as gradient, first principles and second derivatives Quiz: AQA Maths, Unit 7

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson G1, Derivative as gradient, first principles and second derivatives: 20 multiple choice questions for the AQA Maths (7357), Unit 7: Differentiation, written with Revision Ninja.

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The 20 questions

  1. Geometrically, what does the derivative f'(x) represent at a point on a curve?

    • Normal gradient
    • Area under curve
    • Tangent gradient
    • Chord length

  2. Which limit defines the derivative f'(x) from first principles?

    • lim (f(x+h)-f(x))/x
    • lim (f(x+h)-f(x))/h
    • lim (f(x+h)+f(x))/h
    • lim (f(x)-f(h))/h

  3. Using first principles, what is the derivative of x^2?

    • 2x
    • x
    • 2
    • x^2/2

  4. What is the second derivative of y = x^3?

    • 6
    • 6x
    • 3x
    • 3x^2

  5. What does the second derivative f''(x) measure geometrically?

    • Rate of gradient change
    • Gradient of the curve
    • Area under the curve
    • Rate of function change

  6. What changes at a point of inflection on a curve?

    • Y-intercept
    • Domain
    • Gradient sign
    • Concavity

  7. What is the derivative of x^3 with respect to x?

    • 3x
    • x^3/3
    • 3x^2
    • x^2

  8. What is the gradient of y = x^2 at x = 3?

    • 6
    • 9
    • 5
    • 3

  9. If f(x) = x^3, what is f'(2)?

    • 6
    • 8
    • 4
    • 12

  10. Find f''(x) for f(x) = 4x^3 - 2x.

    • 24x
    • 12x^2 - 2
    • 24
    • 12x

  11. What is the gradient of the curve y = x^2 - 4x at x = 1?

    • -4
    • -2
    • 2
    • -3

  12. Using first principles on f(x) = x^2, what does (f(x + h) - f(x))/h simplify to before h tends to 0?

    • x + h
    • 2x - h
    • 2x + h
    • 2x

  13. For f(x) = x^4 - 2x^2, what is f''(x)?

    • 4x^3 - 4x
    • 12x^2 - 2
    • 12x^2 - 4
    • 12x - 4

  14. If f''(x) > 0 on an interval, what shape is the curve there?

    • Convex
    • Discontinuous
    • Concave
    • Linear

  15. If f'(x) is increasing on an interval, what is the sign of f''(x) on that interval?

    • Zero
    • Undefined
    • Negative
    • Positive

  16. Find the second derivative of y = 1/x^2.

    • -2/x^3
    • 2/x^3
    • 6/x^4
    • -6/x^4

  17. A curve has second derivative y'' = 6x - 12. What is the x-coordinate of its point of inflection?

    • 0
    • 2
    • -2
    • 6

  18. Using first principles, find f'(1) for f(x) = 3x^2 + x.

    • 6
    • 7
    • 4
    • 8

  19. Why does f(x) = x^3 have a point of inflection at x = 0?

    • f''(x) is positive
    • f'(x) changes sign
    • f(x) is zero
    • f''(x) changes sign

  20. Estimate the gradient of y = x^2 at x = 1 using the chord from x = 1 to x = 1.01, to 2 decimal places.

    • 2.50
    • 2.10
    • 1.99
    • 2.01

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