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. For f(x) = (x²+1)³, find f'(1).

    Answer: ______________

  2. If g'(a) = 0 and f is differentiable at g(a), then the derivative of f(g(x)) at a is zero.

    Circle one:   True   False

  3. Using radians, f(x) = cos(5x). Find f'(0).

    Answer: ______________

  4. Differentiate (3x²+1)⁵ and identify its inner and outer functions.

    • 30x(3x²+1)⁴; inner 3x²+1; outer u⁵
    • 5(3x²+1)⁴; inner 3x²+1; outer u⁵
    • 30x(3x²+1)⁵; inner 3x²+1; outer u⁵

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

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

  6. To obtain a circle's area rate from A = pi r², substitute its momentary radius as a constant before differentiating in time.

    Circle one:   True   False

  7. A sphere has radius 3 cm and volume rate 72pi cm³/s. With V = (4/3)pi r³, what is its radius rate?

    • 2 cm/s
    • 6 cm/s
    • 8 cm/s

  8. Using radians, what is the derivative of e^(cos(4x))?

    • 4cos(4x)e^(cos(4x))
    • -4sin(4x)e^(cos(4x))
    • -sin(4x)e^(cos(4x))

  9. Using radians, f(x) = e^(sin(x²)). Find f'(0).

    Answer: ______________

  10. A sphere has radius 3 cm increasing at 0.5 cm/s. With V = (4/3)pi r³, what is its volume rate?

    • 9pi cm³/s
    • 18pi cm³/s
    • 4.5pi cm³/s
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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 · The derivative is 6x(x²+1)², giving 24 at 1.
  2. True · The chain rule multiplies the defined outer derivative by g'(a) = 0.
  3. 0 · The derivative is -5sin(5x), which is zero at 0.
  4. 30x(3x²+1)⁴; inner 3x²+1; outer u⁵ · The outer factor is 5(3x²+1)⁴ and the inner derivative is 6x.
  5. 6 metres per second · Multiply the rates: 2(3) = 6, with units (m/m)(m/s) = m/s.
  6. False · Keep r time-dependent while differentiating; substitute its momentary value afterward.
  7. 2 cm/s · The rate equation is 72pi = 36pi dr/dt, giving 2 cm/s.
  8. -4sin(4x)e^(cos(4x)) · Multiply the exponential factor, the cosine derivative and the inner derivative 4.
  9. 0 · The derivative includes the inner factor 2x, which is zero at 0; the other factors are finite.
  10. 18pi cm³/s · The rate equation gives 4pi(3²)(0.5) = 18pi cm³/s.
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