CAIE A-Level Mathematics 3.4.2 Product and Quotient Rules

CAIE A-Level Mathematics 3.4.2 Product and Quotient Rules
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise differentiating algebraic, exponential and trigonometric products or quotients, factorising the result and solving exact stationary-point or proof conditions.

How this is tested

  • apply uv′ + vu′ or (vu′ − uv′)/v² while retaining every chain-rule factor
  • factor or simplify the derivative numerator before setting it equal to zero
  • solve the resulting algebraic or trig condition and substitute back for full coordinates

Question 5(a)

[Maximum number: 4]

A curve has equation y=1+e2x1+3xy=\frac{1+\mathrm{e}^{2 x}}{1+3 x}. The curve has exactly one stationary point P.

Find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} and hence show that the x-coordinate of P satisfies the equation x=16+12e−2xx=\frac{1}{6}+\frac{1}{2} \mathrm{e}^{-2 x}.