CAIE A-Level Mathematics 2.4.2 Product and Quotient Rules

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

Exam points

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

[Maximum number: 3]

Figure for Question 2.4.2—Product and quotient rules question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=\frac{2 \ln x}{3 x+1}$ . The curve crosses the x-axis at the point A and has a maximum point B. The shaded region is bounded by the curve and the lines x=3 and y=0.

Find the gradient of the curve at A.

Differentiate using quotient rule (or product rule)

M1

Obtain $\frac{(3 x+1) \frac{2}{x}-6 \ln x}{(3 x+1)^{2}}$

A1

OE

Substitute x=1 to obtain $\frac{1}{2}$

A1

OE

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2.4.2—Product and quotient rules question 1