ZIMSEC A Level · Pure Mathematics · Form 5, Form 6
Rules
- d/dx(xⁿ) = nxⁿ⁻¹; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x; d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec²x.
- Chain rule: dy/dx = dy/du × du/dx.
- Product rule: d(uv)/dx = u v′ + v u′. Quotient rule: d(u/v)/dx = (v u′ − u v′)/v².
- Implicit differentiation: differentiate each term with respect to x, using d(y²)/dx = 2y dy/dx.
Worked example: Differentiate y = x² sin 3x.
u = x², u′ = 2x; v = sin 3x, v′ = 3 cos 3x
dy/dx = 3x² cos 3x + 2x sin 3x
Applications
- Gradient of a tangent; normal gradient = −1/(tangent gradient).
- Stationary points where dy/dx = 0: d²y/dx² > 0 gives a minimum, < 0 a maximum.
- Rates of change: dA/dt = dA/dr × dr/dt.
Worked example: Find the stationary points of y = x³ − 3x² − 9x + 5 and their nature.
dy/dx = 3x² − 6x − 9 = 3(x − 3)(x + 1) = 0 → x = 3 or x = −1
d²y/dx² = 6x − 6: at x = 3, 12 > 0 (minimum, y = −22); at x = −1, −12 < 0 (maximum, y = 10)
Exam tip: Always give both coordinates of a stationary point and justify its nature.
Check yourself
- Differentiate ln(3x² + 1).
- Differentiate e^(2x)/x.
- The radius of a circle increases at 0.2 cm/s. Find the rate of increase of area when r = 5 cm.
Show answers
- 6x/(3x² + 1)
- e^(2x)(2x − 1)/x²
- dA/dt = 2πr × 0.2 = 2π cm²/s ≈ 6.28 cm²/s
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