CAIE A-Level Mathematics 2.4.2 Product and Quotient Rules

CAIE A-Level Mathematics 2.4.2 Product and Quotient Rules
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise differentiating products and quotients involving algebraic, exponential, logarithmic or trigonometric factors and using the result to locate stationary points.

How this is tested

  • apply uv′ + vu′ to a product and retain chain-rule factors in each differentiated term
  • apply (vu′ − uv′)/v² to a quotient before simplifying its numerator and denominator
  • set the derivative numerator to zero and solve exactly or numerically for stationary points

Question 6

[Maximum number: 4]
Figure for Question 6 — CAIE A-Level Mathematics

The diagram shows the curve with equation y=ln⁡(2x+1)x+3y=\frac{\ln (2 x+1)}{x+3}. The curve has a maximum point M.

Question 6(a)

(a)

Find an expression for dy dx\frac{\mathrm{d} y}{\mathrm{~d} x}.

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Question 6(b)

(b)

Show that the x-coordinate of M satisfies the equation x=x+3ln⁡(2x+1)−0.5x=\frac{x+3}{\ln (2 x+1)}-0.5.

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