Calculus: Derivatives & Optimization
Chapter concept map
Tap a node with a dot to see its explanation.
Key formulas
Power Rule
d/dx[xⁿ] = n · xⁿ⁻¹
Multiply by the exponent, then reduce the exponent by 1.
Chain Rule
d/dx[f(g(x))] = f'(g(x)) · g'(x)
Derivative of the outer function (at the inner) times derivative of the inner.
Log Derivative
d/dx[ln(u)] = u' / u
Derivative of the inside, divided by the inside.
2nd Derivative Test
f''(x) < 0 → local max f''(x) > 0 → local min
Negative curvature = peak. Positive curvature = trough.
Price Elasticity of Demand
ε_D = (dQ/dP) × (P/Q)
Percentage change in quantity demanded per 1% change in price.
- Q
- quantity demanded
- P
- price
Subtopic 1 / 4
The Six Derivative Rules
You are applying to one of Italy's best Finance and Economics master's programmes. The background test just wants to know you can think quantitatively. Let's make derivatives fast.
Six rules. Know them without thinking:
- Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
- Exponential: d/dx[eˣ] = eˣ — and d/dx[e^g(x)] = e^g(x) · g'(x)
- Natural log: d/dx[ln(x)] = 1/x — and d/dx[ln(g(x))] = g'(x)/g(x)
- Product rule: d/dx[f·g] = f'·g + f·g'
- Quotient rule: d/dx[f/g] = (f'g − fg') / g²
- Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
If these aren't fast yet, write them out ten times. Seriously.
Worked example
f(x) = x³ → f'(x) = 3x². f(x) = e^(2x) → f'(x) = 2e^(2x) by chain rule. f(x) = ln(x²) → f'(x) = 2x/x² = 2/x by the log rule.
Exam strategy
Focus on understanding underlying concepts — exam questions test application, not memorization.
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