Topic 5 of 8 · Form 6
Pure 3: further calculus and differential equations
Derivatives of eˣ, ln x, sin, cos, tan; product and quotient rules; implicit and parametric differentiation.
Notes for this topic 1 file, in order
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Differentiation
Derivatives of eˣ, ln x, sin, cos, tan; product and quotient rules; implicit and parametric differentiation.
Worked example: Find dy/dx for x = t², y = 2t³ at t = 2.
dy/dx = (6t²)/(2t) = 3t = 6
Trigonometry
sec, cosec, cot; compound and double angle formulae; R sin(θ ± α) forms.
Integration
Standard integrals including eᵃˣ⁺ᵇ, 1/(ax + b), sin, cos, sec²; integration using trigonometric identities; by substitution; by parts; with partial fractions; the trapezium rule.
Worked example: Find ∫ x cos 2x dx.
u = x, dv = cos 2x → x sin 2x/2 − ∫ sin 2x/2 dx = ½x sin 2x + ¼ cos 2x + c
First-order differential equations
Separable equations; forming equations from rates of change; interpreting solutions.
Worked example: Solve dy/dx = xy² given y = 1 when x = 0.
∫y⁻² dy = ∫x dx → −1/y = x²/2 + c; c = −1 → y = 2/(2 − x²)
Check yourself
- Differentiate e^(2x) sin x.
- Find ∫ tan²x dx.
- Express 3 cos θ + 4 sin θ as R cos(θ − α).
Show answers
- e^(2x)(2 sin x + cos x)
- tan x − x + c
- 5 cos(θ − 53.13°)
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