A-level · Differentiation

Differentiating powers of xExam-style questions

Finding how steep a curve is at any point. What a derivative measures and where it comes from; the power rule and why the power comes down; numbers in front, sums and constants; rewriting roots and fractions as powers; negative and fractional powers; using the gradient function at a point and backwards; and differentiating twice.

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16 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 function $\mathrm{f}$ is defined by

$\mathrm{f}(x) = 4x^3 - 3x^2 + kx + 10$

where $k$ is a constant.
(a) Find, in simplest form,

(i) $\mathrm{f}'(x)$

(ii) $\mathrm{f}''(x)$

(3 marks)

The curve $y = \mathrm{f}'(x)$ and the straight line $y = \mathrm{f}''(x)$ meet at two points. One of these points, $A$ , has $x$ coordinate $3$
(b) Find the value of $k$ .

(2 marks)
(c) Hence find the coordinates of $B$ , the other point where the curve and the line meet.

(3 marks)
Level 4 · 8 marksTry it
2
The curve $C$ has equation

$y = \dfrac{2x + 1}{\sqrt{x}}, \qquad x > 0$

Find the exact coordinates of the stationary point of $C$ . (6 marks)
Level 4 · 6 marksTry it
3
A curve has equation

$y = 4\sqrt{x} - \dfrac{2}{x} \quad \text{for } x > 0$
(a) Find an expression for $\dfrac{dy}{dx}$

(3 marks)
(b) The point $P$ with coordinates $(4, 7.5)$ lies on the curve.

Find an equation of the tangent to the curve at the point $P$

(2 marks)
(c) Show that the curve has no stationary points.

(2 marks)
Level 4 · 7 marksTry it
4
A curve has equation

$y = \dfrac{x^2}{2} + 6\sqrt{x} \quad \text{for } x > 0$
(a) Find an expression for $\dfrac{dy}{dx}$

(3 marks)
(b) The point $P$ with coordinates $(4, 20)$ lies on the curve.

Find an equation of the tangent to the curve at the point $P$

(2 marks)
(c) Show that the curve has no stationary points.

(2 marks)
Level 4 · 7 marksTry it
5
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve $C$ has equation

$y = 2x^2 + 2x\sqrt{x} - 6x \qquad x > 0$
(a) Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ , giving your answer in simplest form.

(2 marks)

The point $P$ lies on $C$ . The tangent to $C$ at $P$ has gradient $16$ and crosses the $y$ -axis at the point $(0,\ k)$ , where $k$ is a constant.
(b) Show that the $x$ coordinate of $P$ satisfies the equation

$4x + 3\sqrt{x} - 22 = 0$

(2 marks)
(c) Hence find the value of $k$ .

(4 marks)
Level 4 · 8 marksTry it