CAIE A-Level Mathematics AS 2.4.2 Product and Quotient Rules Questions

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

Exam points

use vector notation and operations
solve geometric problems with vectors
interpret vector relationships

[Maximum number: 4]

Figure for Question CAIE A-Level Mathematics AS 2.4.2 Product and Quotient Rules Questions question 1 — CAIE A-Level Mathematics AS

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

(a)

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

[ 2 ]

Attempt use of quotient rule
Or equivalent method.

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

(b)

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

[ 2 ]

Equate first derivative to zero and arrange as far as $2 x+1=\ldots$
Confirm $x=\frac{x+3}{\ln (2 x+1)}-0.5$
Answer given - necessary detail needed.

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CAIE A-Level Mathematics AS 2.4.2 Product and Quotient Rules Questions question 1

Question (a)

Question (b)