A-level · Differentiation

Differentiation from first principlesExam-style questions

The definition every differentiation rule is built on. What "the gradient at a point" means, as the limit of chords; the definition f'(x) as a limit; writing out f(x + h) correctly; cancelling h before letting it shrink; setting out the proof the way the marks are given; and finding the gradient at one point from first principles.

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8 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    Given that

    f(x)=2x3−5xf(x) = 2x^3 - 5x,

    use differentiation from first principles to show that f′(x)=6x2−5f'(x) = 6x^2 - 5. (4 marks)
    Level 3 · 4 marksTry it
  2. 2
    Differentiate sin⁡x\sin x with respect to xx from first principles.

    (4 marks)
    Level 3 · 4 marksTry it
  3. 3
    Prove, from first principles, that the derivative of sin⁡x\sin x is cos⁡x\cos x.

    You may assume the addition formula for sin⁡(A+B)\sin(A + B) and the small-angle limits sin⁡hh→1\dfrac{\sin h}{h} \to 1 and cos⁡h−1h→0\dfrac{\cos h - 1}{h} \to 0 as h→0h \to 0. (5 marks)
    Level 3 · 5 marksTry it