MathLesson 1 of 3

Calculus & Optimisation

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 maxf''(x)>0→local min

Part 1 of 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)

Worked example

1f(x) = x³ · ln(2x). u = x³ → u' = 3x². v = ln(2x) → v' = 2/(2x) = 1/x.
2Product rule: f'(x) = 3x²·ln(2x) + x³·(1/x) = 3x²·ln(2x) + x².
3Option A in the document.

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Algebra, Sets & Functions