| layout | page |
|---|---|
| title | Differentiation rules — game-style playbook |
| permalink | /derivative-rules/ |
Plain-language map of the algebra of derivatives: power, sum, product, quotient, and chain. It mirrors the in-game block in Level select → Math tips & snippets / Game → Math concepts (“Differentiation rules — your skill tree”).
| Rule | On the graph | Game metaphor |
|---|---|---|
| Power | (x^n \Rightarrow n x^{n-1}) | “Stage exponent” controls how fast tilt ramps. |
| Constant × | ((kf)' = kf') | One buff scales every local slope. |
| Sum / diff | ((f\pm g)' = f' \pm g') | Stacked modifiers add cleanly. |
| Product | ((uv)' = u'v + uv') | Two meters both contribute. |
| Quotient | (\bigl(\frac{u}{v}\bigr)' = \frac{u'v-uv'}{v^2}) | Top vs bottom track; watch (v=0). |
| Chain | ((f\circ g)' = (f'\circ g),g') | Nested stage: outer × inner slope. |
The derivative curve in-app is still numeric for arbitrary (f); these rules explain closed forms you learn in class and why many stages look related before sampling.
For (y = x^n) (where (x^n) is defined on the interval you care about),
[ \frac{d}{dx} x^n = n, x^{,n-1}. ]
Intuition: the exponent becomes a multiplier out front; the variable’s power drops by one — the slope field “inherits” the degree structure.
[ \frac{d}{dx}\bigl[k,f(x)\bigr] = k,f'(x), \qquad \frac{d}{dx}\bigl[f(x) \pm g(x)\bigr] = f'(x) \pm g'(x). ]
Intuition: scaling or adding functions scales or adds slopes. Linearity is why superposition shows up everywhere in physics and engineering.
[ \frac{d}{dx}\bigl[u(x),v(x)\bigr] = u'(x),v(x) + u(x),v'(x). ]
Mnemonic: “first times derivative of second plus second times derivative of first.” Neither factor can be ignored: both cross-terms matter.
Where (v(x) \neq 0),
[ \frac{d}{dx}\frac{u(x)}{v(x)} = \frac{u'(x),v(x) - u(x),v'(x)}{[v(x)]^2}. ]
Mnemonic: “low (\times) d-high minus high (\times) d-low, over low squared” (with (u) = high, (v) = low). Domain: wherever (v = 0) or you cross a vertical asymptote / cutout, stop — the formula doesn’t apply there.
If (y = f(g(x))), then
[ \frac{dy}{dx} = f'(g(x))\cdot g'(x). ]
Intuition: a small nudge in (x) moves (g) by about (g'(x),\Delta x); then (f) responds to that inner change at rate (f'(g(x))). Total sensitivity = outer × inner. Matches the game’s inner variable (u = k(x-D)): change (x) → change (u) → change (f(u)).
- Exponential / log rules ((e^{kx}), (\ln x)) appear on named stages — see [Math concepts]({% link math-concepts.md %}) and topic pages.
- Implicit differentiation (e.g. circle stage) handles relations that aren’t solved as (y=f(x)) everywhere.
- Linear approx / differentials tie slopes to “nearby value” predictions — useful for bounds and TMUA-style elimination.
- [Math concepts & snippets]({% link math-concepts.md %}) — full glossary and exam crosswalk
- [AP Calculus BC — prep]({% link ap-calculus-bc.md %}) — where these rules sit in the BC toolbox
- [TMUA — calculus]({% link tmua-calculus.md %}) — MCQ fluency with product / quotient / chain
Unofficial study notes — not affiliated with College Board, Cambridge Assessment, or Oxford.