0606.14.4: Differentiate products and quotients of functions — IGCSE Additional Mathematics 0606 past paper questions

Real questions on 0606.14.4: differentiate products and quotients of functions

The newest 18 of 18 written questions. Open one to see it on the real paper.

0606/12 · October/November 2025 · Q12a[4]
12 A curve has equation $y = \frac{e ^{(x^{2})}}{x - 2}$ for $x < 2$ . (a) Find the value of $\frac{d y}{d x}$ when $x = 0$ .See it on the paper →
0606/13 · October/November 2025 · Q2[3]
2 Differentiate $x^{2} e^{3 x}$ with respect to $x$ .See it on the paper →
0606/13 · October/November 2025 · Q3[4]
In this question you may use the values in the table below. | $θ$ radians | $sin θ$ | $cos θ$ | $tan θ$ | | :---: | :---: | :---: | :---:| | $\frac{π}{6}$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{3}{3}$ | | $\frac{π}{3}$ | $\frac{3}{2}$ | $\frac{1}{2}$ | $3$ |...See it on the paper →
0606/22 · October/November 2025 · Q7a[2]
(a) Given that $y = x cos 2 x$ , find $\frac{d y}{d x}$ .See it on the paper →
0606/22 · February/March 2025 · Q6a[3]
It is given that $y = \frac{l n ( 2 x ^{2} + 1 )}{x + 2}$ . (a) Find $\frac{d y}{d x}$ See it on the paper →
0606/22 · February/March 2025 · Q9c[2]
In this question, all lengths are in metres and time is in seconds. A particle, $P$ , moves in a straight line such that $t$ seconds after passing through a fixed point $O$ its displacement, $s$ , is given by $s = 5 ln (2 t + 1) - 5 t$ . (c) Find an expression for the acceleration of $P$ in terms of $t$ .See it on the paper →
0606/13 · October/November 2024 · Q7a[2]
7 (a) Given that $y = x^{3} ln x$ , find $\frac{d y}{d x}$ .See it on the paper →
0606/13 · May/June 2024 · Q8[7]
8 The line $L$ is the normal to the curve $y = 3 (5 x + 6)^{\frac{1}{2}}$ at the point where $x = 2$ . The point $(- 2, k)$ , where $k$ is a constant, lies on $L$ . Find the exact value of $k$ .See it on the paper →
0606/12 · October/November 2022 · Q11a[5]
11 A particle $P$ moves in a straight line such that, $t$ seconds after passing through a fixed point $O$ , its displacement, $s$ metres, is given by $s = \frac{( 2 t + 1 ) ^{\frac{3}{2}}}{t + 1} - 1$ . (a) Show that the velocity of $P$ at time $t$ can be written in the form $\frac{( 2 t + 1 ) ^{\frac{1}{2}}}{( t + 1 ) ^{2}} (a + b t)$ , where $a...See it on the paper →
0606/13 · May/June 2022 · Q7a[4]
A curve has equation $y = \frac{( 2 x + 1 ) ^{2}}{x + 5}$ for $x ⩾ 0$ . (a) Show that $\frac{d y}{d x} = \frac{( 2 x + 1 ) ^{\frac{1}{2}}}{( x + 5 ) ^{2}} (A x + B)$ , where $A$ and $B$ are integers to be found.See it on the paper →
0606/12 · February/March 2022 · Q5a[3]
Variables $x$ and $y$ are such that $y = \frac{l n ( 2 x ^{2} - 3 )}{3 x}$ . (a) Find $\frac{d y}{d x}$ See it on the paper →
0606/13 · October/November 2021 · Q10a[5]
10 A curve with equation $y = f (x)$ is such that $\frac{d ^{2} y}{d x ^{2}} = (2 x + 3)^{- \frac{1}{2}} + 5$ for $x > 0$ . The curve has gradient 10 at the point $(3, \frac{19}{2})$ . (a) Show that, when $x = 11$ , $\frac{d y}{d x} = 52$ .See it on the paper →
0606/13 · May/June 2021 · Q2b(ii)[3]
(ii) Hence find $\int_{0}^{2} x (3 x^{2} + 4)^{- \frac{2}{3}} d x$ .See it on the paper →
0606/22 · May/June 2021 · Q10bi[2]
(b) (i) Differentiate, with respect to $x$ , $y = x sin 4 x$ .See it on the paper →
0606/22 · May/June 2021 · Q10bii[4]
4-mark “Show” question — its maths and diagrams are best seen on the paper.
See it on the paper →
0606/12 · February/March 2021 · Q10a[5]
10 A curve has equation $y = \frac{( 2 x ^{2} + 10 ) ^{\frac{3}{2}}}{x - 1}$ for $x > 1$ . (a) Show that $\frac{d y}{d x}$ can be written in the form $\frac{( 2 x ^{2} + 10 ) ^{\frac{1}{2}}}{( x - 1 ) ^{2}} (A x^{2} + B x + C)$ , where $A$ , $B$ and $C$ are integers.See it on the paper →
0606/13 · October/November 2020 · Q2a[3]
(a) Given that $y = \frac{e ^{2 x - 3}}{x ^{2} + 1}$ , find $\frac{d y}{d x}$ See it on the paper →
0606/13 · October/November 2020 · Q4b[3]
(b) The displacement, $x$ m, of a particle from a fixed point at time $t$ s is given by $x = 6 cos (3 t + \frac{π}{3})$ . Find the acceleration of the particle when $t = \frac{2 π}{3}$ .See it on the paper →

What examiners said

The great majority of candidates obtained a fully correct unsimplified answer, with most attempting differentiation of a quotient rather than a product. Either method was acceptable. Subsequent errors in attempted simplification were condoned as these errors led to a loss of marks in the subsequent parts of the question.
— Examiner report, February/March 2025, 0606/22 Q6(a) (Cambridge)

Most candidates realised that they needed to differentiate again to obtain an expression for the acceleration of the particle. It was intended that candidates make use of the chain rule, with many doing so. Some chose to differentiate using the quotient rule, often with errors in the numerator either through incorrect simplification or incorrect differentiation.
— Examiner report, February/March 2025, 0606/22 Q9(c) (Cambridge)

Most candidates attempted to differentiate the given equation as a quotient. Some errors were made in the differentiation using the chain rule and there was evidence of poor use of brackets by many candidates. It is essential that terms are bracketed correctly as many candidates often obtain incorrect results because of incorrect or missing brackets. There were some attempts at re-writing the equa...
— Examiner report, May/June 2022, 0606/13 Q7(a) (Cambridge)