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. What does the derivative f'(x) represent geometrically?

    • The area under the curve y = f(x) from 0 to x
    • The value of f(x) at the point where x = 0
    • The gradient of the tangent to the curve y = f(x) at the point with x-coordinate x
    • The y-intercept of the curve y = f(x) at each point

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

    • lim as h tends to 0 of (f(x) - f(h))/x
    • lim as h tends to 0 of (f(x + h) - f(x))/h
    • lim as h tends to 0 of (f(x + h) + f(x))/h
    • lim as h tends to 0 of f(x + h) - f(x)

  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 of a function measure?

    • The rate of change of the gradient, which indicates whether the curve is convex or concave
    • The rate of change of the area under the curve
    • The value of the function at its maximum point
    • The gradient of the normal at each point of the curve

  6. What is a point of inflection?

    • A point where the curve crosses the x-axis
    • A point where the function is undefined
    • A point where the gradient of the curve is always zero
    • A point where the curve changes between convex and concave, often where the second derivative changes sign

  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. A curve has f''(x) > 0 for all x in an interval. What can be said about the curve on that interval?

    • It is convex (concave up), so its gradient is increasing
    • Its gradient is zero throughout the interval
    • It is concave down, so its gradient is decreasing
    • It has a maximum point somewhere in the interval

  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. Explain why f(x) = x^3 has a point of inflection at the origin.

    • f''(0) = 1, so the curve is convex at the origin
    • f''(x) = 6 is constant, so there is no point of inflection
    • f''(0) = 3, so the origin is a local minimum
    • f''(x) = 6x changes sign at x = 0, so the concavity changes there

  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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