Differential Calculus

Key concepts

• First principles: f'(x) = lim_{h→0} [f(x+h) − f(x)] / h
• Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
• Tangent at x = a has gradient f'(a)
• Stationary points: f'(x) = 0
• Concavity: f''(x)

How it works

Differentiate using the rule (almost always the power rule in Matric). Set f'(x) = 0 to find stationary points; use f''(x) or a sign table to classify them. For optimisation, write the quantity to optimise as a function of one variable, differentiate, and solve f'(x) = 0.

Step by step

1. Differentiate

Use the power rule term by term.

2. Solve f'(x) = 0

Find x-values of stationary points.

3. Classify

Use f''(x) or sign analysis.

4. Interpret

State coordinates and nature of each point.

Worked examples

Find the stationary points of f(x) = x³ − 3x.

1. f'(x) = 3x² − 3
2. 0 = 3x² − 3 → x = ±1
3. f(1) = −2; f(−1) = 2

Answer: (1, −2) local min; (−1, 2) local max

Find the equation of the tangent to f(x) = x² − 3x at x = 1

1. f(1) = 1 − 3 = −2, so the point is (1; −2)
2. f′(x) = 2x − 3, so the gradient is f′(1) = −1
3. y − (−2) = −1(x − 1)

Answer: y = −x − 1

A rectangle has a perimeter of 60 cm. Find the dimensions that give the maximum area.

1. Constraint: 2x + 2y = 60, so y = 30 − x
2. Area A = x(30 − x) = 30x − x²
3. A′(x) = 30 − 2x = 0, so x = 15
4. A″(x) = −2 < 0, so this is a maximum

Answer: 15 cm by 15 cm, an area of 225 cm²

Use first principles to find f′(x) if f(x) = −x²

1. f(x + h) − f(x) = −(x + h)² + x² = −2xh − h²
2. Divide by h: −2x − h
3. Take the limit as h → 0

Answer: f′(x) = −2x

Common mistakes

• Forgetting to bring down constants when differentiating
• Mixing up local max and min
• Not writing the equation of the tangent in y = mx + c form
• Forgetting to check that a stationary point is a maximum (or minimum) in optimisation

Frequently asked questions

Is first principles tested every year?

Yes — usually a 4–6 mark question early in Paper 1.

What does the second derivative tell me?

f″(x) gives the concavity. At a stationary point, f″(x) < 0 means a local maximum and f″(x) > 0 means a local minimum.