1. Differentiate the following with respect to $x$:
$(3x-7)^{12}$
$(6x+5)^{\frac{5}{3}}$
$36(3x-7)^{11}$
$10(6x+5)^{\frac{2}{3}}$
2. Differentiate the following with respect to $x$:
$\sqrt{5x-3}$
$\dfrac{1}{(3-x)^4}$
$\dfrac{5}{2\sqrt{5x-3}}$
$\dfrac{4}{(3-x)^5}$
3. Differentiate the following with respect to $x$:
$\dfrac{4}{3x^3-x}$
$\sqrt[3]{(2x^3+3)^4}$
$\dfrac{-4(9x^2-1)}{(3x^3-x)^2}$
$8x^2\sqrt[3]{2x^3+3}$
4. Find and simplify the derivatives of the following:
$x\sqrt{x-1}$
$x^2\sqrt{3x+1}$
$\dfrac{3x-2}{2\sqrt{x-1}}$
$\dfrac{x(15x+4)}{2\sqrt{3x+1}}$
5. Fully factorise the derivative of $$2x^4(5+x)^3$$
$2x^3(20+7x)(5+x)^2$
6. Differentiate $\dfrac{x}{x+2}$ using both the product rule and the quotient rule to show that they give the same solution.
$\dfrac{2}{(x+2)^2}$
7. 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.
$4(2x+3)^5(14x+3)$
8. Find the derivative of the following in fully factorised form: $$(2x+\sqrt{4x+1})^5$$
9. 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 $\dfrac{ax+b}{2(x-1)^{\frac{1}{2}}}$
$\dfrac{9x-5}{2(x-1)^{\frac{1}{2}}}$
10. By differentiating a suitable expression, find:
$\displaystyle\int(2x-5)^6\ \mathrm{d}x$
$\displaystyle\int\sqrt{3-x}\ \mathrm{d}x$
$\dfrac{1}{14}(2x-5)^7+c$
$-\dfrac{2}{3}(3-x)^{\frac{3}{2}}+c$
11. By differentiating a suitable expression, find:
13. Find the coordinates of the stationary point(s) on the following curves:
$y=x\sqrt{x+12}$
$y=(1-3x)(3-x)^3$
$(-8,-16)$
$(1,-16)$ and $(3,0)$
14. Find the equation of the normal to the curve $$y=x^2(2-x)^3$$ when $x=1$.
$y=x$
15. Differentiate the following with respect to $x$:
$\dfrac{x^2}{x+4}$
$\dfrac{1-x}{x^3+2}$
$\dfrac{x^2+8x}{(x+4)^2}$
$\dfrac{2x^3-3x^2-2}{(x^3+2)^2}$
16. Differentiate the following with respect to $x$:
$\dfrac{\sqrt{x}}{3x+2}$
$\dfrac{2x+1}{\sqrt{x-3}}$
$\dfrac{2-3x}{2\sqrt{x}(3x+2)^2}$
$\dfrac{2x-13}{2(x-3)^{\frac{3}{2}}}$
17. 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 $40$
18. 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.
$\dfrac{1}{2}$
19. 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 $144\pi$.
$1.5$
20. A closed cylinder has a total surface area of $300\pi$.
Show that its volume is $150\pi r - \pi r^3$
Hence find the maximum volume of the cylinder to 3 significant figures.
See video.
$2220$
21. 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 $2$.
$2$
22. 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 $x=2$.
$3$
23. 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.
$\dfrac{1}{48}$
24. 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.
