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
In this question you must show detailed reasoning.

Figure 1 shows the curve $C$ with equation

$y = x\sqrt{4 - x}, \qquad 0 \leqslant x \leqslant 4$

The region $R$ , shown shaded, is bounded by $C$ and the $x$ -axis.Figure 1
(a) Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{8 - 3x}{2\sqrt{4 - x}}$ . (4 marks)
(b) Find the exact coordinates of the maximum point of $C$ . (3 marks)
(c) Using the substitution $u = 4 - x$ , show that the area of $R$ is $\dfrac{128}{15}$ . (5 marks)
Level 5 · 12 marksTry it
2
Figure 1A football is kicked from a point $O$ on horizontal ground and first hits the ground again at the point $A$ , where $OA = d$ metres. Figure 1 shows the path of the football from $O$ to $A$ .

At a horizontal distance of $x$ metres from $O$ , the height of the football above the ground is $h$ metres. In a model of the kick, the football is treated as a particle moving in a vertical plane and

$h = 1.2x - 0.2x\text{e}^{0.05x} \qquad 0 \leqslant x \leqslant d$

Use the model to answer parts (a),
(b) and (c).
(a) Find the value of $d$ , giving your answer to 2 decimal places.

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

(3 marks)
(b) Show that the football is at its maximum height when

$x = 20\ln\left(\dfrac{120}{20 + x}\right)$

(4 marks)

Using the iteration formula

$x_{n+1} = 20\ln\left(\dfrac{120}{20 + x_n}\right) \qquad x_1 = 25$
(c) (i) find the value of $x_2$ to 2 decimal places,

(ii) find, by repeated iteration, the horizontal distance of the football from $O$ when it is at its maximum height. Give your answer to 2 decimal places.

(3 marks)
Level 5 · 10 marksTry it
3
$\mathrm{f}(x) = (2x - 1)\mathrm{e}^{-x}$
(a) Show that $\mathrm{f}'(x) = (3 - 2x)\mathrm{e}^{-x}$ , and hence find the exact coordinates of the stationary point of the curve $y = \mathrm{f}(x)$ . (5 marks)
(b) Determine the nature of the stationary point. (3 marks)
(c) Find the exact value of $\displaystyle\int_0^1 (2x - 1)\mathrm{e}^{-x}\;\mathrm{d}x$ . (3 marks)
Level 5 · 11 marksTry it
4
The diagram shows the curve with equation $y = \sin x\cos 2x$ for $0 \leqslant x \leqslant \tfrac{1}{4}\pi$ . The curve has a maximum point $M$ .
(a) Find the $x$ -coordinate of $M$ .

(6 marks)
(b) Using the substitution $u = \cos x$ , find the exact area of the shaded region.

(4 marks)
Level 5 · 10 marksTry it