CAIE A-Level Mathematics 3.4.3 Implicit and Parametric Differentiation

CAIE A-Level Mathematics 3.4.3 Implicit and Parametric Differentiation
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise deriving dy/dx from implicit or parametric curves, simplifying it before substitution and using zero, undefined or reciprocal gradients for tangents and normals.

How this is tested

  • attach dy/dx to each differentiated y-term and isolate it from an implicit relation
  • calculate dy/dt and dx/dt before dividing to obtain a simplified parametric gradient
  • use zero or undefined tangent gradients and negative reciprocals to find requested points or lines

Question 6

[Maximum number: 9]

A curve has parametric equations

x=tan⁡θ,y=sin⁡θ−2sin⁡3θ,x=\tan \theta, \quad y=\sin \theta-2 \sin ^{3} \theta,

for 0<θ<12π0<\theta<\frac{1}{2} \pi.

Question 6(a)

(a)

Show that dy dx=6cos⁡5θ−5cos⁡3θ\frac{\mathrm{d} y}{\mathrm{~d} x}=6 \cos ^{5} \theta-5 \cos ^{3} \theta.

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Question 6(b)

(b)

（b）Find the equation of the normal to the curve at the point where it crosses the x－axis．Give your answer in the form y=m x+c ，where m and c are exact constants．

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