courses › Calculus

The rules for rates of change: power, product, chain.

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Opens at level 33.

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skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Power rule par 20 s not tried yet

  worked example

  f(x) = −4x⁴ − 5x³ − 7x² + 9. What is f′(2)?

  Answer: -216

  1. Power rule: bring each power down and lower it by one, so f′(x) = −16x³ − 15x² − 14x.
  2. f′(2) = −16(2)³ − 15(2)² − 14(2) = −128 − 60 − 28 = −216.

• Product rule par 30 s not tried yet

  worked example

  f(x) = (−x² − 1)(4x − 5). What is f′(3)?

  Answer: -82

  1. Product rule: f′ = u′v + uv′ with u = −x² − 1 and v = 4x − 5.
  2. u′ = −2x and v′ = 4.
  3. At x = 3: (−6) × 7 + (−10) × 4 = −42 − 40 = −82.

• Chain rule par 30 s not tried yet

  worked example

  f(x) = e^(x² − 2x − 3). What is f′(−1)?

  Answer: -4

  1. Chain rule: f′(x) = e^(x² − 2x − 3) × (2x − 2).
  2. At x = −1 the exponent is (−1)² − 2(−1) − 3 = 1 + 2 − 3 = 0, so e⁰ = 1.
  3. f′(−1) = 1 × (2(−1) − 2) = −4.

• eˣ, ln, sin, cos par 25 s not tried yet

  worked example

  f(x) = 2e³ˣ. What is f′(0)?

  Answer: 6

  1. Chain rule: the inside 3x has derivative 3, so f′(x) = 2 × 3e³ˣ = 6e³ˣ.
  2. At x = 0, e⁰ = 1, so f′(0) = 6.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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