Differentiation

By drawing a suitable tangent, estimate the gradient of the curve at point $P$.

Feb/March 2019

The curve is given by the equation $y = x^3 - 3x + 4$.

Feb/March 2020

The table lists several values of $y = 2x^3 - 4x^2 + 3$.

Feb/March 2020

The curve is described by $y = x^{3} - 2x^{2} + 5$.

Feb/March 2021

The grid displays the graph of $y = a + bx^2$. The curve passes through the points $(0,4)$ and $(1,1)$.

Feb/March 2021

$f(x) = x^3 - 3x^2 - 4$

Feb/March 2023

Sketch the graph of $y = 4 - 3x$ on the axes.

Feb/March 2024

The curve is defined by $y = x^3 + x^2 - x$. One stationary point occurs at $\left(\dfrac{1}{3}, -\dfrac{5}{27}\right)$.

Feb/March 2025

The graph plots a cyclist’s speed over a $30$ second journey. The vertical axis is labelled Speed (m/s), while the horizontal axis is Time (seconds).

Feb/March 2025

The graph of $y = 10 - 8x^2$ for $-1.5 \leq x \leq 1.5$ is plotted on the grid.

May/June 2018

Find the $x$-coordinates of the points on the graph of $y = x^5 - 5x^4$ at which the gradient is 0.

May/June 2021

Simplify the fraction $\frac{x^2 - 25}{x^2 - x - 20}$.

May/June 2021

Find the value of $a$ for which the equation $y = x^3 - 4x^2 + 4x$ can be written as $y = x(x-a)^2$.

May/June 2021

Find the value of $f(2)$.

May/June 2022

The curve is given by $y = x^3 - kx^2 + 1$. At $x = 2$, its gradient is $6$.

May/June 2022

Find the gradient of the curve at the point $(1, -11)$ on the curve.

May/June 2023

Show that the equation $y = (x - 4)(x + 1)(x - 2)$ can be expressed as $y = x^3 - 5x^2 + 2x + 8$.

May/June 2023

The curve is described by $y = x^4 - 8x^2 + 5$.

May/June 2023

The graph for $y = f(x)$ is plotted on the grid.

May/June 2024

On the coordinate axes, sketch the graph of $y=x^2+7x-18$. Label on your sketch the points where the graph crosses the $x$-axis and the $y$-axis.

May/June 2024

Show that $(-1, reminder: 6)$ is a stationary point of the curve.

May/June 2024

$y = x^7 - 7x^6$

May/June 2024

The diagram displays the graph of $y = 3x - x^3$. It meets the $x$-axis at $A$, $O$ and $B$, and its turning points are $P$ and $Q$.

May/June 2025

Differentiate the expression $x^3 - 3x^2 + 1$.

May/June 2025

The curve $y = x^3 + 2x^2 - 4x$ is displayed on the grid. The diagram includes the $x$-axis and $y$-axis, along with the cubic curve.

Oct/Nov 2016

Complete the value table for $y = \frac{x^3}{3} - x^2 + 1$.

Oct/Nov 2016

The table gives some values for $y = x^3 - 3x - 1$.

Oct/Nov 2018

Suppose $f(x)=\frac{x^2}{4}-\frac{4}{x}$, $x\neq0$.

Oct/Nov 2018

The speed-time graph provides details of a train journey. Speed is given in km/min and time is measured in minutes.

Oct/Nov 2019

Differentiate the expression $6 + 4x - x^{2}$.

Oct/Nov 2020

Find the coordinates for the points $A$, $B$ and $C$.

Oct/Nov 2020

Find the solution to $f(x) = 14$.

Oct/Nov 2020

Factorise the quadratic $24 + 5x - x^2$.

Oct/Nov 2020

Find the coordinates of the turning points on the graph of $y=x^3-12x+6$. Show all your working.

Oct/Nov 2021

A sketch of the graph of $y=f(x)$ for $-1.5 \le x \le 6$ is shown. Five labelled points on the graph of $y=f(x)$ are marked on the diagram: $A(-1.5,7.9)$, $B(-1,7)$, $C(1.2,10)$, $D(5,3)$ and $E(6,5.5)$.

Oct/Nov 2022

All measurements in this question are in centimetres. The diagram illustrates a cuboid solid with a square base. Its vertical height is marked $9 - x$, and each edge of the base is marked $x$. The diagram is labelled NOT TO SCALE.

Oct/Nov 2022

The derivative of $2ax^7 + 3x^k$ is $42x^6 + 15x^{k-1}$.

Oct/Nov 2023

Differentiate the expression $x^3 - 4x^2 - 3x$.

Oct/Nov 2023

The diagram gives a sketch of the graph of $y = 4x^3 - x^4$. It cuts the $x$-axis at the origin $O$ and again at the point $A$. Point $B$ is a maximum point.

Oct/Nov 2023

The diagram contains a sketch of $y=18+5x-2x^2$.

Oct/Nov 2023

A curve is given by the equation $y = x^3 - 9x^2 - 48x$.

Oct/Nov 2024

The sketch represents the graph of $y=3+2x-x^2$. Point $A$ has coordinates $(-1,0)$, and point $B$ has coordinates $(2,3)$.

Oct/Nov 2024

$y=x^3+3x^2-13x$

Oct/Nov 2025