Math / Helper companion / Product & quotient
Uncut · book companion · ~96–103
Two factors changing — not (uv)′ = u′v′
Product & quotient
When the output is a product (or quotient) of two changing pieces, the slope is not “derivative times derivative.” Climb from the prior ability to differentiate each factor: product rule first, then quotient as the matching partner (or as product with 1/v). (produktregeln · kvotregeln)
V1 — Product: both factors can change.
Builds on: ability to form u′ and v′ (intro + trig/chain tools).
Tool gained: (uv)′ = u′v + uv′.
- Principle. If two lengths multiply to an area (or two signals multiply), a tiny change hits either factor — first times rate of second, plus second times rate of first. Not u′v′ alone.
- Model. (uv)′ = u′v + uv′.
- Prediction / Answer. y = x² sin x → y′ = 2x sin x + x² cos x. Transfer: revenue = price × quantity when both drift with time.
V2 — Quotient from the product idea.
Builds on: product rule (V1 tool) — write u/v = u · v⁻¹ or expand the limit.
Tool gained: (u/v)′ = (u′v − uv′)/v² (v ≠ 0).
- Principle. A ratio’s steepness must account for numerator rising and denominator rising (which can pull the ratio down) — same two-factor idea with a minus in the cross term.
- Model. (u/v)′ = (u′v − uv′) / v².
- Prediction / Answer. y = sin x / x (x ≠ 0) → y′ = (x cos x − sin x)/x². Transfer: efficiency = output/input when both change.
1. Poor dump → rewrite
Book dump: boxed (uv)′ and (u/v)′ with no why, and the trap (uv)′ = u′v′ left unchallenged. Our rewrite: two changing factors → sum of “each one’s rate × the other’s level”; quotient keeps the cross-minus over v².
Weapons: (uv)′ = u′v + uv′ → (u/v)′ = (u′v − uv′)/v². Combine with chain when a factor is itself composed.
2. Hands-on — catch the u′v′ trap
For y = x · sin x at a chosen x, compare true product rule vs the fake u′v′.
…
3. Fake → real
Fake
“Differentiate y = x cos x.” Drill with no story.
Real
A sliding panel’s width is x(t) and a light sensor’s response is proportional to cos x. Signal S = x cos x. Peak rate of S needs S′ = cos x − x sin x — product rule as a control limit, not a worksheet line.
Gate — four, all required
1. Reconstruct: (uv)′ equals…
2. Why does quotient build on product (climb, not branch dump)?
3. Predict: y = x² / cos x. y′ numerator over cos² x is…
4. Transfer: price p(t) and sales q(t). Revenue R = pq. R′ is…