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

    Answer: ______________

  4. For y = 6x², what is the gradient at x = 3?

    Answer: ______________

  5. Differentiate y = 5x³ - 8x² + 6.

    • 15x² - 16x
    • 15x² - 16x + 6
    • 5x² - 8x
    • 15x² - 8x

  6. Differentiate y = (x + 3)(x - 2).

    • 1
    • 2x + 1
    • 2x - 6
    • 2x - 1

  7. Written as a power of x, what is the fourth root of x?

    • x⁴
    • x⁻⁴
    • x^(1/4)
    • x^(3/4)

  8. Sam differentiates y = x⁻³ and writes -3x⁻². Where is the slip?

    • the power should have gone down to -4, not up to -2
    • the -3 should not have come down to the front at all
    • a negative power cannot be differentiated by this rule
    • the sign in front should have stayed positive

  9. For y = x³ - 4x² + 5x, what is the gradient at x = 2?

    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. 24 · dy/dx = 6x², and putting x = 2 gives 6 times 4, which is 24.
  4. 36 · dy/dx = 12x, and putting x = 3 gives 12 times 3, which is 36.
  5. 15x² - 16x · 5x³ gives 15x², -8x² gives -16x, and the constant 6 goes to zero. So dy/dx = 15x² - 16x.
  6. 2x + 1 · The rule needs a sum of powers, so multiply out first: (x + 3)(x - 2) = x² + x - 6. Then dy/dx = 2x + 1.
  7. x^(1/4) · A root becomes a fractional power, and the root number goes on the bottom of the fraction, so the fourth root of x is x^(1/4).
  8. the power should have gone down to -4, not up to -2 · One is subtracted from the power, and -3 minus 1 is -4. Sam added one instead, so his answer should be -3x⁻⁴.
  9. 1 · dy/dx = 3x² - 8x + 5. At x = 2 that is 12 - 16 + 5, which is 1.
  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