Math & StatsLesson 1 of 5

Calculus: Derivatives & Optimization

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 maxf''(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

Part 1 of 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

1f(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.

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Why is this the correct answer?

Eliminate any option that contradicts something stated directly in the lesson, then check which one actually follows from the rule you just learned. Tell me which option confused you and I'll point to the exact line.

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If you can explain why the wrong answers are wrong, you know the topic. Remembering the right one only gets you halfway.

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Advanced Calculus: L'Hôpital, Integration & Multivariable