Question

find the derivative of f(x) = 4 / (3x - x^2)

Answer 1

\[f(x)=\frac{ 4 }{ 3x-x^2 }\]

Answer 2

\[\frac{ 4 }{ 3(x+h)-(x+h)^2 }-\frac{ 4 }{ 3x-x^2 }\]

Answer 3

are you using the rules for derivatives or are you still using the limit method?

Answer 4

limit method

Answer 5

ahh its a lot more complicated that way unfortunately give me a sec to write out the work myself to make sure I got it then I'll walk through the process with you

Answer 6

is it just 0, if i use substitution?

Answer 7

wait no, that's just the limit, i need the derivative

Answer 8

It's definitely not zero. If you have an answer you want me to check I can do that a lot quicker.

Answer 9

which limit method are you using there are two.

Answer 10

umm, i didnt learn the one with deltas, but...\[\lim(x>0) \frac{ f(x+h)-f(x) }{ h }\]

Answer 11

are you sure that your formula is not suppose to have h approaching to zero not x?

Answer 12

oh yeah sorry
lim(h->0)

Answer 13

so lets write out that formula subbing in the f(x) part.\[\lim_{h \rightarrow 0}\frac{ \frac{ 4 }{ 3(x+h)-(x+h)^2 }-\frac{ 4 }{ 3x-x^2 } }{ h }\]

Answer 14

rationalize denominators?

Answer 15

yes

Answer 16

\[\frac{ 4(3x-x^2)-4[3(x+h)-(x+h)^2]}{ (3x-x^2)[3(x+h)-(x+h)^2] }\]

Answer 17

the numerator should now be \[\frac{ 4(3x-x^2)-4(3x+3h-x^2-2hx-h^2) }{ (3x-x^2)(3x+3h-x^2-2hx-h^2) }\]

Answer 18

now expand

Answer 19

ok

Answer 20

the numerator I mean we may luck out with the denominator

Answer 21

when i tried the problem before, i got stuck in the denominator algebra

Answer 22

numerator:
\[-12h + 8hx +4h^2\]

Answer 23

in fact we do because we have the numerator as \[\frac{ 4(h^2-3h+2hx) }{ (3x-x^2)(3x+3h-x^2-2hx-h^2 }\]

Answer 24

or should i factor the 4 out?

Answer 25

It doesn't truly matter. Either way works fine. It's more standard to factor it out if there is a common factor to pull out.

Answer 26

now for messy denominator

Answer 27

What is more important is that you recognize we can pull an h out now

Answer 28

DON"T DO THAT

Answer 29

IIt's a giant waste of time

Answer 30

yeah, but I need to distribute 3x-x^2

Answer 31

no you don't. Wait for me to type out the full equation one more time and you will see what I mean.

Answer 32

\[\lim_{h \rightarrow 0}\frac{ \frac{ 4h(h-3-2x) }{ (3x-3x^2)(3x+3h-x^2-2hx-h^2 } }{h }\]

Answer 33

notice the cancellation we can preform now?

Answer 34

The "h" in the very top of the question cancels with the "h" in the very bottom of the question.

Answer 35

I think:well i did it anyways\[9x+9xh-3x^3-6hx^2-3xh^2-3x^3-3hx^2+x^4+2hx^3+h^2x^2\]

Answer 36

so the h on the bottom is really h/1, soo we can multiply the top by 1/h right?

Answer 37

if thats the way you want to think of it sure! The important part is that they cancel.

Answer 38

Since they cancel each other we can stop and ask ourselves if we let h approach zero is there still a problem? Do you think there is?

Answer 39

well, how does the bottom h cancel all the others?

Answer 40

nvm

Answer 41

I guess you see now how the rest of the hs do not cause the denominator to approach zero itself. What is the final answer then? Do try to simplify!

Answer 42

\[\frac{ 2(-3h+2hx+4h^2) }{ 3x-x^2 }\]

Answer 43

because h approaches 0

Answer 44

u can "ignore" h multiplied by any term

Answer 45

\[\frac{ 2 }{ (3x-x^2) }\]

Answer 46

not quite

Answer 47

you are making a mistake in your simplification. Instead of ignoring the h as you put it actually sub in zero everywhere you see the letter h and see how that works for simplification.

Answer 48

\[\frac{ 0 }{ 3x-x^2 }\]

Answer 49

nope here I'll write in the sub for you

Answer 50

\[\frac{ 4(0-3+2x) }{(3x-x^2)(3x+0-x^2-0-0) }\]

Answer 51

I think I switched the signs on the numerator though

Answer 52

crap we seem to have messed up somewhere one sec

Answer 53

nope we are fine! I checked our final answer with a math program on my laptop.

Answer 54

ok thanks!

Answer 55

\[\frac{ 4(2x-3) }{ (3x-x^2)^2 }\]

Answer 56

wait isnt the denominator 2(3x-x^2), and if it is, then we can factor out the 4 to a 2