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

$2x^2 + xy + y^2 = 16$ .
(a) Show that $\dfrac{dy}{dx} = -\dfrac{4x + y}{x + 2y}$ . (4 marks)
(b) The point $P(2, 2)$ lies on $C$ . Find an equation of the tangent to $C$ at $P$ , giving your answer in the form $ax + by + c = 0$ , where $a$ , $b$ and $c$ are integers. (2 marks)
(c) Find a relationship between $x$ and $y$ that holds at any point of $C$ where the tangent is parallel to the $x$ -axis. (1 mark)
Level 4 · 7 marksTry it
2
The curve $C$ has equation

$x^2 + xy + y^2 = 12$

Find the coordinates of the two points on $C$ at which the tangent is parallel to the $x$ -axis. (7 marks)
Level 4 · 7 marksTry it
3
The curve $C$ has equation $xy = 12$ .

Find an equation of the tangent to $C$ at the point $(3, 4)$ , giving your answer in the form $ax + by = c$ with integer coefficients. (4 marks)
Level 4 · 4 marksTry it
4
The curve $C$ has equation

$x^2 + 3xy - y^2 = 3$

The point $P(1, 2)$ lies on $C$ .

Find an equation of the tangent to $C$ at $P$ . (7 marks)
Level 4 · 7 marksTry it
5
A curve has equation

$xe^{2y} + y^2 - 3x^2 = k$

where $k$ is a constant.

A stationary point of the curve has $y$ -coordinate $\ln 3$

Determine the exact value of $k$

(7 marks)
Level 4 · 7 marksTry it

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