## anonymous 3 years ago ive tried to prove it many times but , i still didnt get it.... if dy/dx = -y/x+2y prove that d''y/dx''=8/(x+2y)^3

1. tkhunny

Do you mean $$\dfrac{-y}{x+2y}$$? That is NOT what you have written. Folks in calculus should know about the Order of Operations. Have you considered the Quotient Rule for differentiation?

2. precal

use the quotient rule

3. anonymous

really ? sorry i dont know how to write it in the right way...yes i use that quotient rule , but then i got this$\frac{ -\frac{ dy }{ dx }.x+2y-(2y.\frac{ dy }{ dx }) }{ (x+2y)^2 }$ then what should i do ?

4. tkhunny

You should try that again. The whole numerator is a mess.

5. anonymous

ok..wait

6. anonymous

someby pls help me..i dont get it..... :{

7. tkhunny

Numerator: (x+2y)(-y') - (-y)(1+2y') Look at it very carefully.

8. anonymous

oh yeahh my numerator is really a mess..then what should i do ?

9. anonymous

what should i do ~~~~~ omgg

10. hartnn

even i tried this more than once, and i don't get numerator = 8, is anything else given ?

11. anonymous

the original question .

12. anonymous

idk what im doing -.-

13. anonymous

first question

14. hartnn

ohhh...so, xy+y^2 =4.... then its easy, did you get that numerator first ? next thing will be to simplify it.

15. hartnn

got this : (x+2y)(-y') - (-y)(1+2y') first ?

16. anonymous

yes i got it just now .haha

17. anonymous

then ?

18. hartnn

now use, y'=-y/(x+2y)

19. anonymous

yes im using that..then quotient rule ?

20. hartnn

(x+2y)(-y') - (-y)(1+2y') is the result of numerator AFTER applying quotient rule, now you have to simplify

21. tkhunny

Just for the record, it is ALWAYS more beneficial to show the ORIGINAL problem statement.

22. hartnn

yeah, that would have saved me few minutes...

23. anonymous

to be honest im stucking again

24. hartnn

what u got after substituting y'=-y/(x+2y) in that numerator ?

25. anonymous

$2y-\frac{ 2y^2 }{ x+2y }$ what about this ? for that numerator

26. anonymous

$\frac{ 2xy+2y }{ (x+2y)^3 }$

27. anonymous

this ???????

28. anonymous

then subtitute (0,2) into that equation ?

29. hartnn

its actually this : $$\huge \frac{ 2xy+2y^2 }{ (x+2y)^3 }=\frac{ 2(xy+y^2) }{ (x+2y)^3 }$$ now use xy+y^2 =4 and you are done

30. hartnn

$$2y-\frac{ 2y^2 }{ x+2y }$$ was correct

31. anonymous

$\frac{ 2xy+2y^2}{(x+2y)^3}$

32. anonymous

this ????am i right ??

33. hartnn

yeah, factoring out 2 will give you 2 (xy+y^2) and xy+y^2 as per the question is 4

34. anonymous

ohhhh........i see ur answer........and i totally got it ! thanks a lot @hartnn !!:) really appreciate ^_^

35. hartnn

welcome ^_^

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