Differentiation (Cambridge (CIE) IGCSE Additional Maths): Exam Questions

A curve has equation $y=\ln (5–3x)$ where $x<\frac{5}{3}$. The normal to the curve at the point where $x=–5$ , cuts the $x$-axis, at the point $P$.
Find the equation of the normal and the $x$-coordinate of $P$.

Variables $x$ and $y$ are such that $y=e^{\frac{x}{2}}+x\cos 2x$ , where $x$ is in radians. Use differentiation to find the approximate change in $y$ as $x$ increases from 1 to $1+h$, where $h$ is small.

The tangent to the curve $y=\ln \left(3x^2-4\right)-\frac{x^3}{6}$, at the point where $x=2$, meets the $y$-axis at the point $P$. Find the exact coordinates of $P$.

A container is a circular cylinder, open at one end, with a base radius of $r$ cm and a height of $h$ cm. The volume of the container is 1000 cm³. Given that $r$ and $h$ can vary and that the total outer surface area of the container has a minimum value, find this value.

Find the $x$-coordinates of the stationary points of the curve $y=e^{3x}{(2x+3)}^6$.

A curve has equation $y=\mathrm{f}\left(x\right)$ and has exactly two stationary points. Given that ${\mathrm{f}}^{"}\left(x\right)=4x-7$, $\mathrm{f}\text{'}(0.5)=0\mathrm{and}\mathrm{f}\text{'}\left(3\right)=0$, use the second derivative test to determine the nature of each of the stationary points of this curve.

In this question all lengths are in centimetres.

The diagram shows a solid cuboid with height $h$ and a rectangular base measuring $4x$ by $x$. The volume of the cuboid is $40{\mathrm{cm}}^3$. Given that $x$ and $h$ can vary and that the surface area of the cuboid has a minimum value, find this value.

Find the equation of the tangent to the curve $2y=\tan 2x+7$ at the point where $x=\frac{\mathrm{\pi }}{8}$.
Give your answer in the form $ax-y=\frac{\mathrm{\pi }}{b}+c$, where $a,b$ and $c$ are integers.

This tangent intersects the $x$-axis at $P$ and the $y$-axis at $Q$. Find the length of $PQ$.

$y=x\sqrt{x+2}$ Given that , show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{Ax+B}{2\sqrt{x+2}}$, where $A$ and $B$ are constants.

Find the exact coordinates of the stationary point of the curve $y=x\sqrt{x+2}$.

Determine the nature of this stationary point.

Differentiate $y=\tan (x+4)-3\sin x$ with respect to $x$.

Variables $x$ and $y$ are such that $y=\frac{\ln \left(2x+5\right)}{2e^{3x}}$. Use differentiation to find the approximate change in $y$ as $x$ increases from $1\mathrm{to}1+h$, where $h$ is small.

It is given that $y=\frac{\tan 3x}{\sin x}$.

Find the exact value of $\frac{\mathrm{d}y}{\mathrm{d}x}$ when $x=\frac{\mathrm{\pi }}{3}$.

Hence find the approximate change in $y$ as $x$ increases from $\frac{\mathrm{\pi }}{3}\mathrm{to}\frac{\mathrm{\pi }}{3}+h,$ where $h$ is small.

Given that $x$ is increasing at the rate of 3 units per second, find the corresponding rate of change in $y$ when $x=\frac{\mathrm{\pi }}{3}$ , giving your answer in its simplest surd form.

It is given that $y=\ln (\sin x+3\cos x)\mathrm{for}0<x<\frac{\mathrm{\pi }}{2}$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Find the value of $x$ for which $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{2}$.

Given that $y=\frac{{\mathrm{e}}^{2x-3}}{x^2+1}$, find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Hence, given that $y$ is increasing at the rate of 2 units per second, find the exact rate of change of $x$ when $x=2$.

A sphere of radius $r$ cm and volume $V$ cm³ is increasing in size with time $t$ seconds. The volume increases at a constant rate of 24 cm³ s^-1.

Find the exact rate at which the radius is increasing when the sphere reaches a volume of $\frac{32\pi }{3}$ cm³.

A sector from a circle of radius $r$ has an internal angle of $\theta$ radians, as shown below.

The perimeter of the sector is 4 units and the area of the sector is $A$ square units.

Show that $A=\frac{8\theta }{{\left(2+\theta \right)}^2}$.

Show that $\frac{\mathrm{d}A}{\mathrm{d}\theta }=\frac{p\left(q-\theta \right)}{{\left(2+\theta \right)}^3}$ where $p$ and $q$ are constants to be found.

Find the maximum area of the sector. You must show that this area is a maximum.

A curve is given by $y=\ln \left(1+x^4\right)$.

Use calculus to find the approximate change in $y$ as $x$ increases from $-1$ to $-1+k$ where $k$ is small.

In this question, all lengths are in centimetres.

The diagram shows a cone of base radius $x$, height $y$ and sloping edge $\sqrt{x^2+y^2}$. The volume of the cone is $10\mathrm{\pi }{\mathrm{cm}}^3$.

Show that the curved surface area, $S$, of the cone is given by $S=\frac{\pi \sqrt{x^6+900}}{x}$.

Given that $x$ can vary and that $S$ has a minimum value, find the value of $x$ for which $S$ is a minimum.

Variables $x$ and $y$ are such that $y=\frac{e^{3x}\sin x}{x^2}$ .Use differentiation to find the approximate change in $y$as $x$ increases from $0.5\mathrm{to}0.5+h$, where $h$is small.

In this question all lengths are in centimetres. The volume, $V$, of a cone of height $h$ and base radius $r$ is given by $V=\frac{1}{3}\pi r^{2}h$

The diagram shows a large hollow cone from which a smaller cone of height 180 and base radius 90 has been removed. The remainder has been fitted with a circular base of radius 90 to form a container for water. The depth of water in the container is w and the surface of the water is a circle of radius $R$.

Find an expression for $R$ in terms of $w$ and show that the volume $V$ of the water in the container is given by $V=\frac{\mathrm{\pi }}{12}{\left(w+180\right)}^3-486000\mathrm{\pi }$.

Water is poured into the container at a rate of $10000{\mathrm{cm}}^3{\mathrm{s}}^{-1}$. Find the rate at which the depth of the water is increasing when $w=10$. 

A curve has equation $y=\frac{\ln \left(3x^2-5\right)}{2x+1}\mathrm{for}3x^2>5$

Find the equation of the normal to the curve at the point where $x=\sqrt{2}$.

Find the approximate change in $y$ as $x$ increases from $\sqrt{2}\mathrm{to}\sqrt{2}+h$, where $h$ is small.

The rectangle $ABCDE$ represents a ploughed field where $AB=300\mathrm{m}$ and $AE=400\mathrm{m}$. Joseph needs to walk from $A$ to $D$ in the least possible time. He can walk at $0.9{\mathrm{ms}}^{-1}$ on the ploughed field and at $1.5{\mathrm{ms}}^{-1}$ on any part of the path $BCD$ along the edge of the field. He walks from $A$ to $C$ and then from $C$ to $D$. The distance $BC=x\mathrm{m}$.

Find, in terms of $x$, the total time, $T\mathrm{s}$, Joseph takes for the journey.

Given that $x$ can vary, find the value of $x$ for which $T$ is a minimum and hence find the minimum value of $T$.

A curve has the equation $y={\mathrm{e}}^{\left(4x-x^2\right)}$.

Find and factorise an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Hence find $\int \frac{\left(2-x\right){\mathrm{e}}^{4x}}{{\mathrm{e}}^{x^2}}\mathrm{d}x$.

Show that the second derivative of the curve $y={\mathrm{e}}^{\left(4x-x^2\right)}$ satisfies the relationship

$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=\left(p+qx+r{x}^2\right)y$

where $p$, $q$ and $r$ are constants to be found.