A-level · Differentiation

Implicit differentiationExam-style questions

Finding the gradient of a curve whose equation cannot be rearranged into y = f(x). Why such curves need a new method; differentiating a y-term with the chain rule; products of x and y; collecting dy/dx and why the answer contains both x and y; gradients, tangents and normals at a point; and finding where the tangent is horizontal or vertical.

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

DifficultyAll Level 3 Level 4 Level 5

1
The curve $C$ has equation

$x^2 + 2xy + 3y^2 = 6$
(a) Show that

$\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{x + y}{x + 3y}$

(3 marks)
(b) Hence find the exact coordinates of the two points on $C$ at which the tangent is parallel to the $x$ -axis. (4 marks)
(c) Show that there are exactly two points on $C$ at which $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ is not defined, and find their coordinates. (3 marks)
Level 5 · 10 marksTry it
2
The curve $C$ has equation

$x^2 - xy + y^2 = 12$
(a) Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2x - y}{x - 2y}$ . (3 marks)
(b) Find the coordinates of the two points on $C$ at which the tangent is parallel to the $y$ -axis. (4 marks)
(c) The line with equation $y = x + k$ , where $k$ is a constant, is a tangent to $C$ .

Find the possible values of $k$ . (3 marks)
Level 5 · 10 marksTry it
3
In this question you must show detailed reasoning.

The curve $C$ has equation

$x^2 - 2xy + 2y^2 = 8$

The tangent to $C$ at the point $P$ is parallel to the $x$ -axis, and the tangent to $C$ at the point $Q$ is parallel to the $y$ -axis. Both $P$ and $Q$ lie in the first quadrant.

Determine the exact area of triangle $OPQ$ , where $O$ is the origin. (7 marks)
Level 5 · 7 marksTry it
4
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A curve $C$ has equation

$x^3 + xy^2 - 5x - 2y = 4$
(a) Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$ .

(5 marks)

$P(1, -2)$ is a point on $C$ .
(b) Show that the normal to $C$ at $P$ is the line with equation

$y = 1 - 3x$

(2 marks)
(c) Using algebra, prove that the normal to $C$ at $P$ does not intersect $C$ at any other point.

You must make your reasoning clear.

(5 marks)
Level 5 · 12 marksTry it
5
A river flows along a channel through flat, horizontal land.

The diagram below shows a vertical cross section of the channel. The points $O$ and $P$ mark the two banks of the river at ground level.Kofi proposes the following model for the shape of the cross section

$y^2 + y = a\sqrt{x} - x$

where $a$ is a constant, $x$ metres is the horizontal distance from $O$ , and $y$ metres is the depth of the channel below ground level.
(a) The channel is $9$ metres wide at ground level.

Find the value of $a$

(2 marks)
(b) Show that, according to Kofi's model, the greatest depth of the channel is approximately $1.08$ metres.

(6 marks)
(c) State one limitation of Kofi's model.

(1 mark)
Level 5 · 9 marksTry it

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