knowledge/docs/calculus.txt

CALCULUS FORMULAS AND CONCEPTS

Limits:
- lim(x→a) [f(x) + g(x)] = lim(x→a) f(x) + lim(x→a) g(x)
- lim(x→a) [f(x) × g(x)] = lim(x→a) f(x) × lim(x→a) g(x)
- lim(x→0) (sin x)/x = 1
- lim(x→∞) (1 + 1/x)^x = e

L'Hôpital's Rule:
- For 0/0 or ∞/∞: lim f(x)/g(x) = lim f'(x)/g'(x)

Derivatives:
- Power rule: d/dx(x^n) = nx^(n-1)
- Product rule: d/dx(uv) = u'v + uv'
- Quotient rule: d/dx(u/v) = (u'v - uv')/v²
- Chain rule: d/dx[f(g(x))] = f'(g(x)) × g'(x)

Standard Derivatives:
- d/dx(e^x) = e^x
- d/dx(ln x) = 1/x
- d/dx(sin x) = cos x
- d/dx(cos x) = -sin x
- d/dx(tan x) = sec²x

Optimization:
- Critical points: f'(x) = 0
- Local max: f'(x) = 0 and f''(x) < 0
- Local min: f'(x) = 0 and f''(x) > 0

Common Mistakes:
- Forgetting chain rule
- Sign errors in derivatives
- Not checking endpoints in optimization