Question

Find the derivative of f(x) = 4 divided by x at x = 2.

Answer 1

Rewriting question... Find the derivative of \[f(x) = \frac{4}{x}\] at x = 2.

Answer 2

what have you tried so far?

Answer 3

I thought it would be 2 but 2 isnt part of the answers

Answer 4

i will make a guess that you have to do this by hand, not by the power rule
i.e. you have to compute
\[\lim_{x\to 2}\frac{\frac{4}{x}-2}{x-2}\]

Answer 5

so would I then cancel the x-2 and get 4 as the answer?

Answer 6

heck no

Answer 7

oh

Answer 8

first off i made an assumption that that is what you needed to compute
am i right? there are other ways

Answer 9

i ned to find the derivative at x=2

Answer 10

what method do you have at your disposal? by which i mean
do you have to work from the definition? and if so, which definition
\[f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\] or
\[f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\]

Answer 11

the second one is what is in my leson i believe

Answer 12

ok then since \(a=2\) and \(f(a)=f(2)=\frac{4}{2}=2\) you have to compute \[\lim_{x\to 2}\frac{\frac{4}{x}-2}{x-2}\]

Answer 13

subtract in the numerator first

Answer 14

1?

Answer 15

then cancel

Answer 16

are you guessing?
you are actually pretty close

Answer 17

No 2-2=1 and the bottom is 2-2=1

Answer 18

you need to do some algebra here
if you plug in 2 you get \(\frac{0}{0}\)
need to "simplify" that complex fraction first

Answer 19

btw \(2-2\neq 1\)

Answer 20

oh wow sory 0

Answer 21

I was working on another problem and switched numbers

Answer 22

it is time for algebra
that is what is needed

Answer 23

no wonder my other answer was wrong

Answer 24

but if it 0/0 where do I go then?

Answer 25

@satellite73

Answer 26

subtract

Answer 27

it is algebra
you have a compound fraction, you need to simplify it
subtract in the numerator
then you will have a nice cancellation
then you can plug in 2 for \(x\)

Answer 28

in other words, first compute
\[\frac{4}{x}-2\]

Answer 29

How am I suposed to subtract them, i dont know how

Answer 30

1?

Answer 31

4-2=2/2=1? Is that what you mean @satellite73

Answer 32

what happened to your variable?

Answer 33

your answer should be an expression with an \(x\) in it

Answer 34

I do not know how to do it.

Answer 35

I switched the x for 2. Isnt that what you said to do? @satellite73

Answer 36

subtract \[\frac{4}{x}-2\] your answer needs to have an \(x\) in it

Answer 37

2x?

Answer 38

i don't mean to torture you, if you are not up the the subtraction i will show you have to do it

Answer 39

I am just really confused, in some questions I would just have to substitute the x and here it is totally different.@satellite73

Answer 40

no it is algebra is all
that is the biggest cause of confusion
the actual ideas are not that much
\[\frac{4}{5}-2=\frac{4-5\times 2}{5}\]

Answer 41

similarly
\[\frac{4}{x}-2=\frac{4-2x}{x}\]

Answer 42

Then substitute with 2? So 0/2=0

Answer 43

Or would you cancel the two x and get 2

Answer 44

no that was just the numerator

Answer 45

now you have
\[\frac{\frac{4-2x}{x}}{x-2}\]

Answer 46

which is
\[\frac{4-2x}{x(x-2)}\]

Answer 47

now you can cancel

Answer 48

I keep geting 0 for the answer. Im literaly losing my mind .Not even kiding :/

Answer 49

ok lets back up a second and see what the goal is

Answer 50

you want to replace x by 2, but you cannot

Answer 51

if you do, you keep getting \(\frac{0}{0}\)

Answer 52

yes

Answer 53

that means using algebra, you are going to be able to factor and cancel
it always works this way

Answer 54

eventually you will not get \(\frac{0}{0}\) but you will have to cancel first

Answer 55

since you want to put \(x=2\) and you cannot, what you are gong to cancel is a factor of \(x-2\) in the top and bottom

Answer 56

that will allow you to replace \(x\) by 2
but we haven't gotten there yet!
we need the algebra first

Answer 57

i will write it out

Answer 58

Thankyou

Answer 59

\[\frac{\frac{4}{x}-2}{x-2}=\frac{\frac{4-2x}{x}}{x-2}\] by subtracting in the numerator
\[=\frac{4-2x}{x(x-2)}\] by getting rid of the compound fraction
\[=\frac{2(2-x)}{x(x-2)}\] by factoring \[=-\frac{2}{x}\] since \(\frac{2-x}{x-2}=1\)

Answer 60

now you can replace \(x\) by \(2\) in \(-\frac{2}{x}\) and get \(-\frac{2}{2}=-1\)

Answer 61

two things to note
once is in a week you can do this on your head
you will say the derivative of \(\frac{4}{x}\) is \(-\frac{4}{x^2}\) and if you plug in 2 you get -1, but of course now you can plug in anything you like, not just 2

Answer 62

the second thing to not is all the difficultly is with the algebra
this is where the rubber hits the road
if your algebra is weak you will have a very very hard time
the calculus part is very easy, the computations are what cause problems

Answer 63

Oh okay so whenever I get a problem like that square the bottom and make it negative and I wil get the answer?

Answer 64

heck no

Answer 65

Then forget what I said

Answer 66

you have to do what you have to do
you will learn short cut rules for finding the derivative, but math does not work like "if i see this i do that" never does

Answer 67

To be honest I have ben doing math straight for eight hours and Im done

Answer 68

good!
take a break, have a beer
forget it for the night

Answer 69

Haha If only Thanks for your help though!

Answer 70

yw