Q1
Answer
Differentiate the following with respect to $x$:
  1. $(3x-7)^{12}$
  2. $(6x+5)^{\frac{5}{3}}$
  1. $36(3x-7)^{11}$
  2. $10(6x+5)^{\frac{2}{3}}$
Q2
Answer
Differentiate the following with respect to $x$:
  1. $\sqrt{5x-3}$
  2. $\dfrac{1}{(3-x)^4}$
  1. $\dfrac{5}{2\sqrt{5x-3}}$
  2. $\dfrac{4}{(3-x)^5}$
Q3
Answer
Differentiate the following with respect to $x$:
  1. $\dfrac{4}{3x^3-x}$
  2. $\sqrt[3]{(2x^3+3)^4}$
  1. $\dfrac{-4(9x^2-1)}{(3x^3-x)^2}$
  2. $8x^2\sqrt[3]{2x^3+3}$
Q4
Answer
Find and simplify the derivatives of the following:
  1. $x\sqrt{x-1}$
  2. $x^2\sqrt{3x+1}$
  1. $\dfrac{3x-2}{2\sqrt{x-1}}$
  2. $\dfrac{x(15x+4)}{2\sqrt{3x+1}}$
Q5
Answer
Fully factorise the derivative of $$2x^4(5+x)^3$$
$2x^3(20+7x)(5+x)^2$
Q6
Answer
Differentiate $\dfrac{x}{x+2}$ using both the product rule and the quotient rule to show that they give the same solution.
$\dfrac{2}{(x+2)^2}$
Q7
Answer
Use the substitution $u = 2x+3$ to differentiate $$4x(2x+3)^6$$ with respect to $x$ without using the product rule. Write your answer in fully factorised form.
$4(2x+3)^5(14x+3)$
Q8
Answer
Find the derivative of the following in fully factorised form: $$(2x+\sqrt{4x+1})^5$$
$10\left(1+\dfrac{1}{\sqrt{4x+1}}\right)(2x+\sqrt{4x+1})^4$
Q9
Answer
Use a suitable substitution to differentiate $$y=(3x+1)\sqrt{x-1}$$ with respect to $x$ without using the product rule. Write your answer in the form $\dfrac{ax+b}{2(x-1)^{\frac{1}{2}}}$
$\dfrac{9x-5}{2(x-1)^{\frac{1}{2}}}$
Q10
Answer
By differentiating a suitable expression, find:
  1. $\displaystyle\int(2x-5)^6\ \mathrm{d}x$
  2. $\displaystyle\int\sqrt{3-x}\ \mathrm{d}x$
  1. $\dfrac{1}{14}(2x-5)^7+c$
  2. $-\dfrac{2}{3}(3-x)^{\frac{3}{2}}+c$
Q11
Answer
By differentiating a suitable expression, find:
  1. $\displaystyle\int(5x+1)^{\frac{3}{4}}\ \mathrm{d}x$
  2. $\displaystyle\int\dfrac{2}{(4-3x)^3}\ \mathrm{d}x$
  1. $\dfrac{4}{35}(5x+1)^{\frac{7}{4}}+c$
  2. $\dfrac{1}{3}(4-3x)^{-2}+c$
Q12
Answer
By differentiating a suitable expression, find:
  1. $\displaystyle\int x\sqrt{2x^2-4}\ \mathrm{d}x$
  2. $\displaystyle\int\dfrac{x^2}{3(x^3-5)^4}\ \mathrm{d}x$
  1. $\dfrac{1}{6}(2x^2-4)^{\frac{3}{2}}+c$
  2. $-\dfrac{1}{27}(x^3-5)^{-3}+c$
Q13
Answer
Find the coordinates of the stationary point(s) on the following curves:
  1. $y=x\sqrt{x+12}$
  2. $y=(1-3x)(3-x)^3$
  1. $(-8,-16)$
  2. $(1,-16)$ and $(3,0)$
Q14
Answer
Find the equation of the normal to the curve $$y=x^2(2-x)^3$$ when $x=1$.
$y=x$
Q15
Answer
Differentiate the following with respect to $x$:
  1. $\dfrac{x^2}{x+4}$
  2. $\dfrac{1-x}{x^3+2}$
  1. $\dfrac{x^2+8x}{(x+4)^2}$
  2. $\dfrac{2x^3-3x^2-2}{(x^3+2)^2}$
Q16
Answer
Differentiate the following with respect to $x$:
  1. $\dfrac{\sqrt{x}}{3x+2}$
  2. $\dfrac{2x+1}{\sqrt{x-3}}$
  1. $\dfrac{2-3x}{2\sqrt{x}(3x+2)^2}$
  2. $\dfrac{2x-13}{2(x-3)^{\frac{3}{2}}}$
Q17
Answer
A rectangular garden is fenced on three sides, and the house forms the fourth side. The total length of the fence is 80 metres. The area of the garden is a maximum for this length of fence. Find the dimensions of the garden.
$20$ by $40$
Q18
Answer
The surface area of a sphere, $A = 4\pi r^2$, is increasing at a constant rate of $20\pi$ cm2s-1. Find the rate at which the radius is increasing when the radius is equal to $5$ cm.
$\dfrac{1}{2}$
Q19
Answer
The area of a circle is increasing at a constant rate of $36\pi$ cm2s-1. Find the rate at which the radius is increasing when the area is $144\pi$.
$1.5$
Q20
Answer
A closed cylinder has a total surface area of $300\pi$.
  1. Show that its volume is $150\pi r - \pi r^3$
  2. Hence find the maximum volume of the cylinder to 3 significant figures.
  1. See video.
  2. $2220$
Q21
Answer
The volume $V$ of a shape with height $h$ is increasing at a constant rate of $5$ and is given by $$V = 3 - \sqrt{2h^3+h+7}$$ Find the rate at which the height is decreasing, when the height is $2$.
$2$
Q22
Answer
Given $$y = \dfrac{x^2(4-x)}{4}$$ and that the rate of change of $y$ over time is $3$, find the rate of change of $x$ over time when $x=2$.
$3$
Q23
Answer
A liquid pours into an empty container at $30$ cm3s-1. The volume of the liquid is related to its height in the container by $$V = 24h^2$$ Find the rate at which the height of the liquid is increasing after $12$ minutes in cm per second.
$\dfrac{1}{48}$
Q24
Answer
The volume of water in a container is given by $$V = \sqrt{3h^2+2h^3+8}$$ where $h$ is the height of water. The volume is increasing at $18$ cm3s-1. Find the rate at which the height of the water is increasing when it has a height of $2$ cm.
$6$