The Power Rule for Differentiating Polynomials · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The Power Rule for Differentiating Polynomials

Mathematics · Calculus & Analysis · ages 17-18
Name ______________________   Date ____________

  1. Differentiate y = 5x² + 3x.

    • 5x + 3
    • 10x² + 3
    • 10x
    • 10x + 3

  2. The derivative of the constant 9 is 0.

    Circle one:   True   False

  3. The power rule is a shortcut through the first-principles calculation, so a whole-power case could always be proved by hand.

    Circle one:   True   False

  4. For y = 2x³, what is the gradient at x = 2?

    Answer: ______________

  5. For y = 6/x, what is the gradient at x = 2?

    Answer: ______________

  6. Differentiate y = x⁻⁴.

    • -4x⁻³
    • 4x⁻⁵
    • -4x³
    • -4x⁻⁵

  7. Differentiating x⁻⁶ gives -6x⁻⁵.

    Circle one:   True   False

  8. How should 1/x^2 be rewritten as a power before differentiating?

    • x to the power 2
    • x to the power -2
    • x to the power 1/2
    • 2 times x

  9. For y = 6√x, what is the gradient at x = 16?

    Answer: ______________

  10. Ana differentiates y = 3x² and writes 6x². Where is the slip?

    • the power has to drop by one as well, so it is 6x
    • the 3 should have been left alone, giving 2x²
    • the power should have gone up to 3
    • a coefficient cannot be multiplied by the power
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Answer key

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

The Power Rule for Differentiating Polynomials W1-mt_l7f4j44bR3-s1

  1. 10x + 3 · A sum is differentiated term by term. 5x² gives 10x, and 3x gives 3, so dy/dx = 10x + 3.
  2. True · The graph of y = 9 is a flat horizontal line. It has no steepness anywhere, so its derivative is 0 for every x.
  3. True · The rule was read off results that were derived, not assumed. Doing the definition on x⁵ would give 5x⁴ the long way round.
  4. 24 · dy/dx = 6x², and putting x = 2 gives 6 times 4, which is 24.
  5. -1.5 · Rewrite as 6x⁻¹, so dy/dx = -6x⁻², which is -6/x². At x = 2 that is -6 divided by 4, which is -1.5.
  6. -4x⁻⁵ · Bring -4 down to the front, then take one off the power: -4 - 1 is -5, so dy/dx = -4x⁻⁵.
  7. False · The -6 does come down, but one is then subtracted from the power. -6 minus 1 is -7, so the answer is -6x⁻⁷.
  8. x to the power -2 · 1/x^2 means x raised to -2, which the power rule can handle.
  9. 0.75 · Rewrite as 6x^(1/2). Bringing 1/2 down gives 3, and the power becomes -1/2, so dy/dx = 3x^(-1/2), which is 3/√x. At x = 16 that is 3 divided by 4, or 0.75.
  10. the power has to drop by one as well, so it is 6x · Ana did half the rule. Bringing the 2 down to give 6 is right, but the power then has to fall from 2 to 1, so the answer is 6x.
Worksheet · LightMySky