A-level · Differentiation

The product ruleExam-style questions

Differentiating two functions multiplied together. Why multiplying the two derivatives fails; the growing rectangle behind the rule; the product rule u'v + uv' and how to set it out; a bracket or a root as one factor, with the chain rule inside; factorising the answer; and finding the stationary points of a product from its factors.

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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
Figure 1 shows the curve $C$ with equation

$y = x^2\sqrt{3 - 2x}, \qquad x < \dfrac32$ Figure 1
(a) Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{x(6 - 5x)}{\sqrt{3 - 2x}}$ . (4 marks)
(b) Find the exact coordinates of the stationary points of $C$ . (3 marks)
Level 4 · 7 marksTry it
2
In a model of a sprint race, the speed of an athlete, $v$ m s $^{-1}$ , $t$ seconds after the start of the race, is modelled by the equation

$v = \frac{1}{2}(18 - t)\ln(t + 1) \qquad 0 \leqslant t \leqslant 18$

Using the model,
(a) show that the speed of the athlete is a maximum when $t$ satisfies the equation

$(t + 1)\ln(t + 1) + t - 18 = 0$

(4 marks)

The equation in part (a) has exactly one root, $\alpha$ , in the interval $0 \leqslant t \leqslant 18$
(b) Show that $\alpha$ lies in the interval $[5, 6]$

(2 marks)
(c) Using $t_0 = 6$ as a first approximation for $\alpha$ , apply the Newton–Raphson method once to find a second approximation for $\alpha$ , giving your answer to 3 decimal places.

(2 marks)
Level 4 · 8 marksTry it
3
The function f is defined for all real values of $x$ by $f(x) = (x^2 - 2x + 4)e^{3x}$ .
(a) Find $f'(x)$ .

(3 marks)
(b) Hence show that f is an increasing function.

(4 marks)
Level 4 · 7 marksTry it
4
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of the curve with equation $y = \text{f}(x)$ , where

$\text{f}(x) = \cos^2 x\,\text{e}^{2\sin x} \qquad 0 \leqslant x \leqslant \pi$

The curve touches the $x$ -axis at the point $R$ .
(a) Write down the coordinates of $R$ .

(1 mark)

Figure 1 also shows the two maximum points of the curve, $P$ and $Q$ .
(b) Show that the $x$ coordinates of $P$ and $Q$ satisfy the equation

$\sin^2 x + \sin x - 1 = 0$

(4 marks)
(c) Hence find the $x$ coordinates of $P$ and $Q$ , giving each answer to 3 decimal places.

(2 marks)
Level 4 · 7 marksTry it
5
The curve $C$ has equation

$y = (2x + 1)\cos 2x \qquad 0 \leqslant x \leqslant \dfrac{\pi}{4}$

The point $P$ is the only stationary point on $C$ .
(a) Show, using calculus, that the $x$ coordinate of $P$ is a solution of the equation

$x = \dfrac{1}{2}\arctan\left(\dfrac{1}{2x + 1}\right)$

(4 marks)

The iteration formula

$x_{n+1} = \dfrac{1}{2}\arctan\left(\dfrac{1}{2x_n + 1}\right) \qquad x_1 = 0.3$

is used to find an approximation for the $x$ coordinate of $P$ .
(b) Find, to 4 decimal places, the value of

(i) $x_2$

(ii) $x_3$

(3 marks)
(c) Using a suitable interval and a suitable function which should be stated, show that the $x$ coordinate of $P$ is $0.284$ correct to 3 decimal places.

(2 marks)
Level 4 · 9 marksTry it

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