A-level · Differentiation

The quotient ruleExam-style questions

Differentiating one function divided by another. Why differentiating the top and bottom separately fails; where the rule comes from; the quotient rule (u'v − uv') ÷ v² with the order that matters; simplifying the top and leaving the bottom alone; stationary points and increasing intervals from the top's sign; and when rewriting is quicker than the rule.

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27 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The curve $C$ has equation

$y = \dfrac{\ln x}{x}, \qquad x > 0.$
(a) Determine the exact coordinates of the stationary point of $C$ , and show that it is a maximum point. (5 marks)
(b) The equation

$\ln x = kx,$

where $k$ is a constant, has exactly two distinct real roots. Determine the set of possible values of $k$ , justifying your answer fully. (4 marks)
Level 5 · 9 marksTry it
2
In this question you must show detailed reasoning.

The curve $C$ has equation $y = \dfrac{x - 1}{x^2 + 3}$ .
(a) Write down the coordinates of the points at which $C$ meets the coordinate axes.

(2 marks)

$C$ meets the $y$ -axis at $A$ , and $B$ is the point on $C$ with $x$ -coordinate $-3$ .
(b) Show that $AB$ is parallel to the $x$ -axis.

(2 marks)
(c) Determine the coordinates of each turning point of $C$ .

(4 marks)
(d) Determine the nature of each turning point of $C$ .

(5 marks)
Level 5 · 13 marksTry it
3
Figure 4 shows part of the curve $C$ with equation

$y = \dfrac{\ln x}{x^2}, \qquad x > 0$ Figure 4
(a) Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - 2\ln x}{x^3}$ . (3 marks)
(b) Find the exact coordinates of the stationary point of $C$ , and determine its nature. (5 marks)
(c) Find the exact $x$ -coordinate of the point of inflection of $C$ . (2 marks)
Level 5 · 10 marksTry it
4
A curve has equation $y = \dfrac{x^{2}}{1 + \mathrm{e}^{x}}$ . The curve has a minimum point at the origin and a maximum point $M$ .
(a) Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ and show that the $x$ -coordinate of $M$ satisfies the equation $x = 2 + 2\mathrm{e}^{-x}$ .

(4 marks)
(b) Show by calculation that the $x$ -coordinate of $M$ lies between $2.2$ and $2.3$ .

(2 marks)
(c) Use an iterative formula based on the equation in part (a) to find the $x$ -coordinate of $M$ correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

(3 marks)
Level 5 · 9 marksTry it
5
Figure 1Figure 1 shows a sketch of part of the curve $C$ with equation $y = \text{f}(x)$ , where

$\text{f}(x) = \dfrac{x^2 + 6x + 1}{x + 1} \qquad x \geqslant 0$
(a) Find $\text{f}'(x)$ , writing your answer as a single fraction in its simplest form.

(3 marks)

The point $P(1, 4)$ lies on $C$ . The line $l$ is the normal to $C$ at $P$ .
(b) Show that $l$ has equation $x + 2y = 9$

(3 marks)
(c) Express $\text{f}(x)$ in the form $px + q + \dfrac{r}{x + 1}$ , where $p$ , $q$ and $r$ are integers to be found.

(3 marks)

The finite region $R$ , shown shaded in Figure 1, is bounded by $C$ , $l$ and the coordinate axes.
(d) Use algebraic integration to find the exact area of $R$ , giving your answer in the form $a + b\ln 2$ , where $a$ and $b$ are rational numbers.

(Solutions relying entirely on calculator technology are not acceptable.)

(5 marks)
Level 5 · 14 marksTry it