Cheatsheet 04 - Differentiation
Mathematics for Master Students
The Derivative
The derivative of \(f\) at \(a\) is the instantaneous rate of change, the limit of the difference quotient:
\[f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\]
- Geometric reading: the slope of the tangent line at \((a, f(a))\).
- Business reading: how fast the output reacts to a tiny nudge at \(a\).
- Sign of \(f'\) is a trend report: \(f' > 0\) increasing, \(f' < 0\) decreasing.
- No derivative at kinks (e.g. \(|x|\) at \(0\)) or jumps: differentiable means a smooth graph.
Notation (all the same thing): \(\;f'(x) = \dfrac{df}{dx} = \dfrac{d}{dx}f(x)\).
Differentiation Rules
Building blocks. Memorise these:
| \(f(x)\) | \(f'(x)\) | \(f(x)\) | \(f'(x)\) | |
|---|---|---|---|---|
| \(c\) | \(0\) | \(e^x\) | \(e^x\) | |
| \(x^n\) | \(n x^{n-1}\) | \(a^x\) | \(a^x \ln a\) | |
| \(\sqrt{x} = x^{1/2}\) | \(\frac{1}{2\sqrt{x}}\) | \(\ln x\) | \(\frac{1}{x}\) |
Combination rules. Decompose, then assemble:
| Rule | Formula |
|---|---|
| Constant multiple | \((c f)' = c f'\) |
| Sum | \((f + g)' = f' + g'\) |
| Product | \((f g)' = f'g + f g'\) |
| Quotient | \(\left(\dfrac{f}{g}\right)' = \dfrac{f'g - f g'}{g^2}\) |
| Chain | \(\left(f(g(x))\right)' = f'(g(x)) \cdot g'(x)\) |
The power rule takes every exponent. Rewrite negatives and roots as powers first:
- \(\dfrac{1}{x} = x^{-1} \;\Rightarrow\; \left(\dfrac{1}{x}\right)' = -x^{-2} = -\dfrac{1}{x^2}\)
- \(\sqrt{x} = x^{1/2} \;\Rightarrow\; \left(\sqrt{x}\right)' = \dfrac{1}{2}x^{-1/2} = \dfrac{1}{2\sqrt{x}}\)
- Why: multiplying adds exponents, \(x^{-1} \cdot x = x^0 = 1\) and \(x^{1/2} \cdot x^{1/2} = x\).
Chain rule with \(e\) and \(\ln\): \(\;\left(e^{kx}\right)' = k\,e^{kx}\), and \(\;\left(\ln(g(x))\right)' = \dfrac{g'(x)}{g(x)}\).
Before differentiating, ask: sum, product, quotient or chain? The right rule follows automatically. The chain rule’s inner derivative is not optional: \(\left((3x+2)^5\right)' = 5(3x+2)^4 \cdot 3\), never just \(5(3x+2)^4\).
Tangent Line
The tangent to \(f\) at \(x = a\) has slope \(f'(a)\) and passes through \((a, f(a))\):
\[y = f(a) + f'(a)(x - a)\]
A horizontal tangent (\(f'(a) = 0\)) marks a peak, a valley, or a flat spot, the starting point for optimisation.
Second Derivative & Curvature
Differentiate twice: \(\;f''(x) = \left(f'\right)'(x) = \dfrac{d^2 f}{dx^2}\). While \(f'\) reports how fast \(f\) changes, \(f''\) reports how fast the slope changes.
| Condition on an interval | Shape | Chord vs. graph |
|---|---|---|
| \(f''(x) \geq 0\) | Convex (bends up) | chord on or above |
| \(f''(x) \leq 0\) | Concave (bends down) | chord on or below |
- \(f(x) = x^2\): \(f'' = 2 > 0\), convex everywhere.
- \(f(x) = \ln x\): \(f'' = -\frac{1}{x^2} < 0\), concave: diminishing returns.
- Where \(f''\) changes sign, the bend switches: an inflection point.
Marginal Analysis
Marginal cost is \(C'(q)\), the instantaneous rate at which cost grows at output \(q\):
- Practical reading: approximately the cost of one more unit (the \((q{+}1)\)-th).
- Units: € per unit (a rate, not a total). \(C'(50) = 60\) means the 51st unit costs about €60, not that 50 units cost €60.
- Marginal revenue \(\tilde R'(q)\): approximately the revenue from one more unit sold. Producing more pays off while \(\tilde R'(q) > C'(q)\) (roastery, counted in bags: \(\tilde R(q) = 60q - \frac{q^2}{20}\), \(\tilde R'(q) = 60 - \frac{q}{10}\)).
Common Pitfalls
- \((f \cdot g)' \neq f' \cdot g'\): use the product rule \(f'g + fg'\).
- Chain rule: never forget the inner derivative. Multiply by the derivative of the inside.
- Rewrite \(\frac{1}{x}\) and \(\sqrt{x}\) as powers before applying the power rule.
- \(C'(q)\) is a rate in € per unit, not a total cost in €.