Math / Helper companion / Exp & log
Uncut · book companion · ~100–107
Self-slope, inverse slope, light growth
Exp, ln & rate ∝ amount
Exponential growth is the curve whose slope matches its own height. Climb from prior derivative tools: earn (eˣ)′ = eˣ, then (ln x)′ via inverse/chain, then a light reading of y′ = ky as “rate proportional to amount” — not a full DE chapter leap. (exponentialfunktionen · naturliga logaritmen · differentialekvation)
V1 — The curve that matches its own slope.
Builds on: derivative as slope function (intro) — applied to a new family.
Tool gained: (eˣ)′ = eˣ; with chain, (e^{kx})′ = k e^{kx}.
- Principle. There is a smooth positive curve whose steepness at each height equals that height — that self-matching property names eˣ as the natural exponential for calculus.
- Model. d/dx [eˣ] = eˣ. Then d/dx [e^{kx}] = k e^{kx} (chain from trig companion).
- Prediction / Answer. At x = 0, eˣ = 1 and slope = 1. For y = e^{0.2t}, y′ = 0.2 e^{0.2t} — 20% of current height per unit time.
V2 — Ln as inverse → slope 1/x.
Builds on: (eˣ)′ = eˣ (V1) + chain / inverse-function slope idea.
Tool gained: (ln x)′ = 1/x for x > 0; (ln|g|)′ = g′/g when defined.
- Principle. Ln undoes exp. If y = ln x then eʸ = x; differentiate both sides with chain and solve for y′.
- Model. d/dx [ln x] = 1/x (x > 0).
- Prediction / Answer. d/dx [ln(3x)] = 3/(3x) = 1/x. Transfer: relative rate y′/y becomes a difference of logs’ slopes in log-plots.
V3 — Light DE: rate proportional to amount.
Builds on: (e^{kx})′ = k e^{kx} (V1 + chain). Not a full DE course.
Tool gained: y′ = ky has solutions y = C e^{kt}; read k as relative growth rate.
- Principle. “Grows in proportion to how much is already there” means the slope tool equals a constant times the height — exactly what e^{kt} satisfies.
- Model. If y′ = ky, then y = C e^{kt} fits. (Existence/uniqueness deep theory stays Soon.)
- Prediction / Answer. Colony with y′ = 0.1 y and y(0) = 200 → y = 200 e^{0.1t}. At t = 10, y ≈ 200 e ≈ 544 — same chain tool, growth story.
1. Poor dump → rewrite
Book dump: “memorize (eˣ)′ and (ln x)′” then a wall of DE recipes. Our rewrite: eˣ earns itself as self-slope; ln follows by undoing; y′=ky is one reading of (e^{kt})′ — full DE methods stay a later branch.
Weapons: (eˣ)′=eˣ → (ln x)′=1/x → y′=ky ↔ C e^{kt}. Next climb: spend f′ on graphs and max/min.
2. Hands-on — relative rate k
y = e^{kt}. Compare height and slope at t = 0 (both start related by k).
…
3. Fake → real
Fake
“Differentiate y = e^{3x}.” Symbol drill.
Real
A cooling/heating deviation model uses ΔT = 5 e^{−0.3t}. The rate of approach to ambient is ΔT′ = −0.3 · 5 e^{−0.3t}. You size a sensor sample rate from peak |ΔT′| — chain on exp, now a lab job.
Gate — four, all required
1. Reconstruct: why is (eˣ)′ = eˣ special?
2. (ln x)′ = 1/x builds on…
3. Predict: if y′ = 2y and y(0) = 3, then y(t) = ?
4. Climb honesty: full DE methods after this page?