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

The chain rule: rates times rates

Mathematics · Calculus & Analysis · ages 18-19
Name ______________________   Date ____________

  1. Let f(x) = (x squared + 1)^3. Find f'(1).

    Answer: ______________

  2. To differentiate a composition of three functions, you apply the chain rule two times.

    Circle one:   True   False

  3. Let f(x) = (x² + 1)³. Find f'(1).

    Answer: ______________

  4. What is the derivative of (3x squared + 1)^5, and which are the layers?

    • 30x(3x squared + 1)^4 with inner 3x squared + 1 and outer u^5
    • 5(3x squared + 1)^4 with the same layers
    • 30x(3x squared + 1)^5 with the same layers

  5. y is in metres, x is in metres and t is in seconds. With dy/dx = 2 and dx/dt = 3, what is dy/dt?

    • 6 metres per second
    • 6 metres
    • 6 seconds per metre

  6. In a related rates problem, you should substitute the known values into the equation before differentiating.

    Circle one:   True   False

  7. A spherical balloon is being inflated. Its radius is 3 cm and growing at 0.5 cm per second. How fast is the volume growing right then? (Volume of a sphere: V = (4/3)πr³)

    • 18π cm³ per second
    • 9π cm³ per second
    • 4.5π cm³ per second

  8. What is the derivative of sin(ln(x²)) for x > 0?

    • cos(ln(x²)) · 2/x
    • cos(ln(x²)) · 1/x²
    • cos(ln(x²)) · 2x

  9. Let f(x) = e^(sin(x squared)). Find f'(0).

    Answer: ______________

  10. A balloon has radius 3 cm growing at 0.5 cm per second. How fast does its volume grow (V = (4/3) pi r cubed)?

    • 9 pi cm cubed per second
    • 18 pi cm cubed per second
    • 4.5 pi cm cubed per second
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Answer key

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

The chain rule: rates times rates W1-mt_kdhl4dmwJn-s1

  1. 24 · f' = 6x(x squared + 1) squared, and at x = 1 that is 6 times 4 = 24.
  2. True · Each extra layer adds one more multiplication by an inner derivative, so three layers means two uses of the chain rule.
  3. 24 · f'(x) = 3(x² + 1)² · 2x, and plugging in x = 1 gives 3 · 4 · 2 = 24.
  4. 30x(3x squared + 1)^4 with inner 3x squared + 1 and outer u^5 · Outside power rule first, then multiply by the inner derivative 6x.
  5. 6 metres per second · Multiply rates and units: (m/m) times (m/s) gives m/s, and 2 times 3 = 6.
  6. False · You differentiate first to get an equation relating the rates, then substitute the specific values. Substituting early freezes the quantities and gives the wrong derivative.
  7. 18π cm³ per second · dV/dt = 4πr² · dr/dt. With r = 3 and dr/dt = 0.5, that is 4π · 9 · 0.5 = 18π cm³ per second.
  8. cos(ln(x²)) · 2/x · Two layers need the chain rule twice: the derivative of sin is cos of the inside, and the derivative of ln(x²) is 2x/x² = 2/x.
  9. 0 · The derivative carries a 2x factor, which is 0 at x = 0.
  10. 18 pi cm cubed per second · dV/dt = 4 pi r squared times dr/dt = 4 pi times 9 times 0.5.
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