Calculus & Optimisation
Chapter concept map
Tap a node with a dot to see its explanation.
Key formulas
Power Rule
d/dx[xⁿ] = n · xⁿ⁻¹
Chain Rule
d/dx[f(g(x))] = f'(g(x)) · g'(x)
Product Rule
d/dx[u·v] = u'v + uv'
Log Derivative
d/dx[ln(u)] = u' / u
L'Hôpital
lim f/g = lim f'/g' (when 0/0 or ∞/∞)
Power Integral
∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ −1)
2nd Derivative Test
f''(x) < 0 → local max f''(x) > 0 → local min
Subtopic 1 / 3
The Five Derivative Rules
Five rules. Know them without hesitation.
- Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
- Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
- Product rule: d/dx[u·v] = u'v + uv'
- Log rule: d/dx[ln u] = u'/u
- Exponential: d/dx[eˣ] = eˣ; d/dx[e^g(x)] = e^g(x) · g'(x)
⚡ Bocconi combines two rules in one question. The classic is f(x) = x³ · ln(2x) — product rule applied to functions involving the log rule.
Worked example
f(x) = x³ · ln(2x). u = x³ → u' = 3x². v = ln(2x) → v' = 2/(2x) = 1/x. Product rule: f'(x) = 3x²·ln(2x) + x³·(1/x) = 3x²·ln(2x) + x². Option A in the document.
Exam strategy
Focus on understanding underlying concepts — exam questions test application, not memorization.
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