Differentiation from First Principles · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Differentiation from First Principles

Mathematics · Calculus & Analysis · ages 16-17
Name ______________________   Date ____________

  1. What does (x + h)² expand to?

    • x² + h²
    • x² + xh + h²
    • x² + 2xh + h²
    • x² + 2h

  2. You may set h = 0 first and cancel the h in the denominator afterwards.

    Circle one:   True   False

  3. For f(x) = x³, first principles gives f'(x) = 3x². What is f'(2)?

    Answer: ______________

  4. For f(x) = x^2 at x = 2, the difference quotient simplifies to 4 + h. What value does it approach as h gets tiny?

    Answer: ______________

  5. Zara starts on f(x) = x² and writes the top of the fraction as x² + h² - x². Where is the slip?

    • squaring the bracket also gives a middle term, and it has been dropped
    • she should have written x² + 2h + h² for the bracket
    • the h² should have been h
    • the whole top should have been divided by h before the bracket was squared, not after

  6. Differentiate f(x) = x² + 5x from first principles. What is f'(x)?

    • 2x + 5x
    • 2x + 5
    • x² + 5
    • 2x + 5 + h

  7. Cancelling the h in (2xh + h²) / h is only allowed because h is not zero.

    Circle one:   True   False

  8. Differentiating f(x) = x^3 from first principles gives which result?

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

  9. Differentiate f(x) = 2x³ from first principles. What is f'(x)?

    • 3x²
    • 6x
    • 2x²
    • 6x²

  10. Ben divides (3x²h + 3xh² + h³) by h and writes 3x² + 3xh² + h³. Where is the slip?

    • every term loses exactly one h, and his last two still carry one too many
    • the corrected line is 3x + 3xh + h², since the x loses a power too
    • the h³ cancels completely and should have gone
    • he should have factorised the h out before dividing, which changes the final answer
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Differentiation from First Principles W1-mt_pdfPztEjw_-s1

  1. x² + 2xh + h² · Multiplying out (x + h)(x + h) gives x² + xh + hx + h², and the two middle terms collect into 2xh.
  2. False · Setting h = 0 first leaves 0 / 0, which has no value, and there is no longer anything to cancel. The cancelling has to come first.
  3. 12 · Put 2 in for x: 3 times 2² is 3 times 4, which is 12.
  4. 4 · At x = 2 the quotient simplifies to 4 + h, which closes in on 4 as h gets tiny.
  5. squaring the bracket also gives a middle term, and it has been dropped · Squaring a bracket is not squaring each term. The middle term 2xh is the one that survives to become the answer, so losing it leaves h² / h, which gives 0 instead of 2x.
  6. 2x + 5 · f(x + h) = (x + h)² + 5(x + h) = x² + 2xh + h² + 5x + 5h. Subtracting f(x) leaves 2xh + h² + 5h, and dividing by h gives 2x + h + 5, which becomes 2x + 5.
  7. True · Dividing top and bottom by h needs h to be something other than zero. A real chord always has its two points apart, so h is never zero while the fraction is being simplified.
  8. 3x^2 · The h terms cancel and the limit leaves 3x^2.
  9. 6x² · f(x + h) = 2(x³ + 3x²h + 3xh² + h³). Subtracting 2x³ leaves 6x²h + 6xh² + 2h³, and dividing by h gives 6x² + 6xh + 2h², which becomes 6x².
  10. every term loses exactly one h, and his last two still carry one too many · Dividing a sum by h divides every term. Ben did the first one and copied the other two unchanged, so his last two terms still carry one h too many.
Worksheet · LightMySky