Overview
This chapter applies the derivative to real problems: rates of change, tangents/normals, increasing–decreasing behaviour, and finding maxima and minima.
1. Rate of Change
dy/dx measures the rate of change of y with respect to x. Related rates link two changing quantities (e.g. dV/dt to dr/dt).
2. Tangents and Normals
Slope of tangent at (x₁,y₁) = m = dy/dx |₍x₁₎
Tangent: y − y₁ = m(x − x₁)
Normal: y − y₁ = (−1/m)(x − x₁)
3. Increasing / Decreasing and Extrema
f'(x) > 0 → increasing ; f'(x) < 0 → decreasing
Critical points: f'(x) = 0
Second derivative test: f''(x) < 0 → maximum ; f''(x) > 0 → minimum
4. Mean Value Theorems
- Rolle's theorem: if f(a) = f(b), there is c with f'(c) = 0.
- Lagrange's MVT: f'(c) = (f(b) − f(a))/(b − a).
Worked Examples
Example 1 — Rate of change
The radius of a circle increases at 2 cm/s. Find the rate of increase of area when r = 5 cm.
A = πr² → dA/dt = 2πr (dr/dt) = 2π(5)(2) = 20π cm²/s
Example 2 — Tangent
Find the tangent to y = x² at (2, 4).
dy/dx = 2x = 4 at x = 2
Tangent: y − 4 = 4(x − 2) → y = 4x − 4
Example 3 — Maxima/minima
Find the maximum/minimum of f(x) = x³ − 3x + 2.
f'(x) = 3x² − 3 = 0 → x = ±1
f''(x) = 6x : f''(1) = 6 > 0 → minimum at x=1, f(1)=0
f''(−1) = −6 < 0 → maximum at x=−1, f(−1)=4
Example 4 — Lagrange's MVT
Verify MVT for f(x) = x² on [1, 3].
f'(c) = (f(3) − f(1))/(3 − 1) = (9 − 1)/2 = 4
f'(x) = 2x → 2c = 4 → c = 2 ∈ (1, 3) ✓
Common Mistakes
- Using f'(x) sign test incorrectly for increasing/decreasing.
- Forgetting the second-derivative test to classify extrema.
- Slope of normal is −1/m, not −m.
- Not checking that c lies inside (a, b) in MVT problems.
Exam Tips & Revision Checklist
- For optimisation: write the quantity, differentiate, set f'=0, then test.
- Tangent uses dy/dx; normal uses −1/(dy/dx).
- Practised: related rates, tangent line, maxima/minima, MVT.
Frequently Asked Questions
Find f'(x) = 0 (critical points). If f''(x) < 0 it is a maximum; if f''(x) > 0 it is a minimum.