Differentiation rules > Quotient rule

Exercise 1

Differentiate the following functions.

a

$f\left(x\right)=\frac{x+1}{{x}^{2}-16x}$

b

$y\left(x\right)=\frac{1}{{x}^{2}-4x+5}$

c

$H\left(t\right)=\frac{\sqrt{2t+6}}{3t}$

d

$GTK\left(q\right)=\frac{2{q}^{3}-10{q}^{2}+60q+120}{q}$

e

$f\left(x\right)=\frac{2x}{{x}^{2}-10}$

f

$y\left(x\right)=\frac{-4}{1-3{x}^{2}}$

g

$A\left(r\right)=\frac{2r}{\sqrt{4r+8}}$

h

$GO\left(p\right)=200p+400+\frac{2000}{p}$

Exercise 2

Here you see part of the graph of the function $f$. The function rule is $f\left(x\right)=\frac{8x+12}{{x}^{2}+4}$.

a

Algebraically calculate the extrema of $f$.

b

Solve: $f\left(x\right)<\frac{3}{2}$.

c

The graph of $f$ intersects the $x$ -axis in $A$ and the $y$ -axis in $B$. Show that the line $AB$ is tangent to the graph of $f$.

Exercise 3

Some company's wrapping department has been ordered to make bar shaped boxes of which the length is four times the width. Every box is decorated with two silk ribbons as shown in the picture. The volume of the box should be $1$ liter. The company wants to minimize the use of ribbon.

a

Compose a formula for the length $L$ of the needed ribbon as a function of the width $x$ of the box.

b

Use differentiation to calculate for which dimensions of the box the amount of ribbon is as small as possible. Give your answer in millimeters.

Exercise 4

Here you see part of the graph of $f\left(x\right)=\frac{{(x+3)}^{3}}{3{x}^{2}}$ with $-5\le x\le 10$ .

a

Show that $f\prime \left(x\right)=\frac{(x-6){(x+3)}^{2}}{3{x}^{3}}$.

b

Calculate the (local) minimum of $f$.

c

Why is the point $(-3,0)$ an inflection point of the graph of $f$?

Exercise 5

A direct current circuit consists of a $12$ Volt battery with an internal resistance of $12$ ohm and a variable resistance of $R$(ohm). The power $P$(in Watt) generated by this circuit is given by $P=R{I}^{2}$. The current $I$ is given by $I=\frac{12}{R+12}$.

a

Express the generated power in $R$, the variable resistance.

b

Calculate the maximum of the generated power using differentiation.