## anonymous 5 years ago how do you find the log of a square root?

1. anonymous

log sqrt(x) = log x^(1/2) = (1/2)logx

2. anonymous

ok, I have the log of the square root of x^2 +2 ?????

3. anonymous

ok, then log sqrt(x^2 + 2) = log (x^2 + 2)^(1/2) = (1/2)log (x^2 + 2)

4. anonymous

do you know how to find the derivative of it?

5. anonymous

yes, use the following, d/dx(logu) = 1/ulna

6. anonymous

let u = x^2 + 2, then y = logu^(1/2) = (1/2)logu = (1/2)/ulna = (1/2)/(x^2 + 2)lna

7. anonymous

u did not specify base a however

8. anonymous

its the application of the chain rule to the derivative of a logarithm with base a , provided a > 0

9. anonymous

i do not even know what you mean by that. the whole problem is to differentaite x^5 divided by (1-10x)(sqrt ofx^2 +2)

10. anonymous

ok so u want me to differentiate x^5/(1-10x)(sqrt(x^2 + 2))?

11. anonymous

srry, i thought u wanted me to to differentiate log sqrt(x^2 + 2) my bad

12. anonymous

yes but I need to know specifically how to do the expression involving the square root. I cannot understand that; maybe it's the algebra involved to figure it out? /can you do the whole problem, focusing on how you get the derivative of the square root?

13. anonymous

sure :)

14. anonymous

ur my last client for this morning, i gotta go shower after this

15. anonymous

well I sure am glad I found you! Thanks!

16. anonymous

ok since, this is an ugly problem for application of the quotient rule, we r going to let y equal the expression and take the natural log of both sides

17. anonymous

go on, pls

18. anonymous

are you still there?

19. amistre64

whats the question?

20. anonymous

I need to find the derivative of x^5/(1-10X)(sqrtx^2+2) using logs.

21. anonymous

My problem is the expression with the sqrt in it. the others I am comfortable with finding, but that sqrt is killing me!

22. amistre64

$\frac{x^5}{1-10x}*\sqrt{x^2 +2}$ is this the equation? or is that sqrt stuck under the bar?

23. anonymous

the square root is stuck under the bar next to the (1-10x)

24. amistre64

got it..... and why do we have to use logs to do this? is that in the directions? or something you thought might help?

25. anonymous

it was in the instructions to do so.

26. amistre64

odd instructions :)

27. anonymous

are they? I am so new to this, 2 weeks into calc and I think the objective was teach us logs and how to differentiate them using all the forms of expressions possible.

28. amistre64

$\frac{\log(x^5)}{\log[(1-10x)(\sqrt{x^2+2})]}$

29. anonymous

yes!

30. amistre64

anything with an exponent gets dragged to the fron....like this: 5 log(x) ..... that takes care of the top right? or at least is a step we can take..now for th e bottom

31. amistre64

log(ab) = log(a) + log(b) sooooo..... log(1-10x) + log[sqrt(x^2 +2)] right so far?

32. anonymous

yes

33. amistre64

do you recall that radicals such as squaree roots and cube roots and the like are fraction exponentes? sqrt(4) = 4^(1/2) = 2 right?

34. anonymous

yes

35. amistre64

good; and after all that work i see a mistale that my brain made lol.... we first need to take the log of the whole entire fraction; not portions of it :)

36. amistre64

then we can split it up lol

37. amistre64

$\log(\frac{a}{b*c}) = \log(a)-[\log(b)+\log(c)]$ like this :)

38. anonymous

I trust you! go on! brain mistakes are allowed! ; )

39. amistre64

$5 \log(x) - [\log(1-10x) + \frac{1}{2}\log(x^2+2)]$

40. amistre64

$5 \log(x) - \log(1-10x) - (1/2)\log(x^2+2)$

41. amistre64

thats as basic as you can get it; now take the derivatives :) and the "log" part doesnt have to be "log" it can also be "ln"; the natural log...ln migh tmake life easier with derivatives

42. amistre64

$Dx[5 \ln(x) - \ln(1-10x) - \frac{1}{2}\ln(x^2+2)]$

43. amistre64

5/x - 1/(1-10x) - 1/(x^2+2)

44. amistre64

the derivative of ln(x) = Dx/x

45. amistre64

that second term should be: -10/(1+10x) then :)

46. amistre64

.....ack!!! +10/(1-10x) ill get it right eventually lol

47. anonymous

youre great! Pls don't apologize!

48. amistre64

did it make sense what I did :)

49. anonymous

so far it makes sinse except where you said that it was a +10. I thought the whole equation was the expression -another expression-another expression. Why the +?

50. amistre64

the middle term in logs is : - ln(1-10x) the derivative of ln(....) is: the derivative of (....) derivative of (1-10x) ------------------ = ------------------ (....) (1-10x) the second term the becomes.... dont forget the initial (-) in fron of it -10x - ------ = +10/(1-10x) :) (1-10x)

51. anonymous

ok it's the "negative minus a negative gives a positive" thing, right?

52. amistre64

exactly :)

53. anonymous

ok, let's move to the sqrt thing...I cannot stand the suspense!

54. amistre64

2 negative always gives a positive; -5(-3) = 15 7--3 = 10 -6/-3 = 2 -6 -2 = -8....except for that one lol

55. anonymous

; )

56. amistre64

the sqrt became the thrid term: - 1/2 ln(x^2+2) right? which derives to... 1 2x - -- ------- = -x/(x^2+2) right? 2 (x^2+2)

57. anonymous

no how did you get that?

58. amistre64

tell me the last part that makes sense to you and i can unravel the mystery from there :)

59. anonymous

i understand where you got the third term -1/2 ln(x^2+2), but finding the derivative of it is where I am baffled.

60. amistre64

i had to switch to firefox, IE acts wierd on this site

61. amistre64

ok.... Do we have to derive the -1/2 part? or can we pull it aside and leave it alone?... visually that is.

62. amistre64

D(5a^2) = 5 D(a^2) = 5 (2a) = 10a right?

63. anonymous

leave it alone visually is fine. I understand that it has to be put back in when we are done.

64. amistre64

good; then lets derive the ln(x^2+2) part :) D(x^2 +2) --------- right? so all we really have to do is derive the top (x^2 +2)

65. anonymous

OMG! I get it!!!!!!!!!!!!!!!!! Can I try to explain it you so I know for sure?

66. amistre64

you can try :)

67. anonymous

Ok, give me a minute to write it all down then type it to you, but I am going to work only with the sqrt expression since that was the one giving me trouble. Can you hold for like 2 minutes?

68. amistre64

illl be around :)

69. anonymous

ok, here goes. i will do my best with the ^ sign and stuff, ok? Are yo with me? Here goes:::::: d/dx(ln(x^2+2)^1/2) = -1/2(ln(x^2+2)) = -1/2(1/x^2+2)d/dx(x^2+2) = -1/2(1/x^2+2)(2x) = -2x/2(x^2+2) = (after canceling out the 2) -x/(x^2+2) Am I right? Is that the derivative of the natural log of the sqrt of x^2+2?

70. amistre64

that is very good :) except for the spurious (-) sign which is just a result of it being the third term subtracted from the others... it looks like you got a handle on it :)

71. anonymous

should the 1/2 be then positive or negative?

72. amistre64

it should be positive since if it were standing all alone by itself: Dx[ln(x^2+2)^(1/2))] = Dx[(1/2) ln(x^2+2)] = 1/2 Dx(ln(x^2+2)) 1/2 2x/x^2+2 the 2s cancel to give you... x/(x^2 + 2) :)

73. anonymous

thank you, thank you, thank you!!! You helped me when no one else could make me understand! Can I give yuou a medal for this?

74. amistre64

you can if you want ;)

75. anonymous

how do I do that exactly?

76. amistre64

you night have to press your refresh button on your browser...its usually just the f5 button on te keyboard. That will refresh the page and allow you to see a "give medal" option next to my name :)

77. anonymous

ok, IO will do that now, although I agree that IE is weird with this; the page kept scrolling up and down on its own! i like Firefox so much better, but MathXL only works with IE.

78. amistre64

is MathXL a program you are working with?

79. amistre64

did the refresh work out for you?