Derivatives
The rules for rates of change: power, product, chain.
Pen and paper is fine · no calculator needed why?
Opens at level 33. You're level 1. You can read and practice here now.
the idea
The derivative f′(x) measures how fast f(x) changes as x changes: the slope of its graph at that x. So f′(2) is the slope at x = 2.
A few rules build every derivative here. The power rule handles each term of a polynomial, the product rule two factors multiplied, and the chain rule one function inside another. Four derivatives are worth knowing by heart: eˣ stays eˣ, ln x becomes 1/x, sin x becomes cos x, and cos x becomes −sin x.
techniques
Power rule, term by term
- For each term, multiply by the power and lower the power by one: 5x³ becomes 15x².
- A term like 4x becomes 4, and a plain number becomes 0.
- Put the number into f′, not into f.
worked example
f(x) = 2x³ − 5x² + 4x − 1. What is f′(2)?
- f′(x) = 6x² − 10x + 4.
- f′(2) = 6 × 4 − 10 × 2 + 4 = 24 − 20 + 4 = 8.
Answer: 8
Product rule
- Call the factors u and v, and find u′ and v′.
- f′ = u′v + uv′: change one factor at a time, then add the two parts.
- Put the number into all four pieces, then combine.
- At x = 0: e⁰ = 1, sin(0) = 0, cos(0) = 1.
worked example
f(x) = (x + 2)(x² − 3). What is f′(1)?
- u = x + 2, u′ = 1. v = x² − 3, v′ = 2x.
- At x = 1: u is 3, v is −2, u′ is 1, v′ is 2.
- f′(1) = 1 × (−2) + 3 × 2 = −2 + 6 = 4.
Answer: 4
Chain rule: outside, then inside
- Differentiate the outside function and leave the inside alone.
- Multiply by the derivative of the inside.
- (inside)ⁿ gives n(inside)ⁿ⁻¹ × inside′, and e^(inside) gives e^(inside) × inside′.
- ln(inside) gives inside′ ÷ inside. sin(kx) gives k·cos(kx), and cos(kx) gives −k·sin(kx).
worked example
f(x) = (3x − 4)³. What is f′(2)?
- Outside: 3(3x − 4)². Inside derivative: 3.
- f′(x) = 3(3x − 4)² × 3.
- At x = 2 the inside is 2, so f′(2) = 3 × 4 × 3 = 36.
Answer: 36
watch out for
- Putting the number into f instead of f′. Differentiate first, then substitute.
- Multiplying the two derivatives for a product. The product rule adds two parts, u′v and uv′.
- Forgetting the inside derivative: (2x + 1)⁴ needs an extra factor 2, and ln(x² + 1) needs 2x on top.
- Sign slips with cos: its derivative is −sin, and cos π is −1.
practice
Power rule
worked example
f(x) = −4x⁴ − 5x³ − 7x² + 9. What is f′(2)?
Answer: -216
- Power rule: bring each power down and lower it by one, so f′(x) = −16x³ − 15x² − 14x.
- f′(2) = −16(2)³ − 15(2)² − 14(2) = −128 − 60 − 28 = −216.
Product rule
worked example
f(x) = (−x² − 1)(4x − 5). What is f′(3)?
Answer: -82
- Product rule: f′ = u′v + uv′ with u = −x² − 1 and v = 4x − 5.
- u′ = −2x and v′ = 4.
- At x = 3: (−6) × 7 + (−10) × 4 = −42 − 40 = −82.
Chain rule
worked example
f(x) = e^(x² − 2x − 3). What is f′(−1)?
Answer: -4
- Chain rule: f′(x) = e^(x² − 2x − 3) × (2x − 2).
- At x = −1 the exponent is (−1)² − 2(−1) − 3 = 1 + 2 − 3 = 0, so e⁰ = 1.
- f′(−1) = 1 × (2(−1) − 2) = −4.
eˣ, ln, sin, cos
worked example
f(x) = 2e³ˣ. What is f′(0)?
Answer: 6
- Chain rule: the inside 3x has derivative 3, so f′(x) = 2 × 3e³ˣ = 6e³ˣ.
- At x = 0, e⁰ = 1, so f′(0) = 6.