## ChrisV 3 years ago Find the derivative of the function. y=x/sqrt(x^2+1)

1. nanda082

use uv rule

2. ChrisV

$y=x/\sqrt{x ^{2}+1}$

3. ChrisV

well I understand part of it

4. ChrisV

use quotient rule

U-substitution, yo. Or whatever they call it nowadays.

6. ChrisV

then gemeraal power rule

7. ChrisV

general

Oh, wait, derivative, not integral. XD

9. ChrisV

10. cuddlepony

y=x/sqrt(x^2+1) Use Quenient and Chain rule

Alright, product rule of x*(x^2+1)^(-1/2), (x^2+1)^(-1/2)-(1/2)x(2x)(x^2+1)^(-3/2) Yeah.

12. ChrisV

$y'=(x ^{2}+1)^{1/2} - x ^{2}(x ^{2}+1)^{-1/2}/x ^{2}+1$

13. ChrisV

thqats the derivative now i have to figure out how to simplify it

14. cuddlepony

15. cuddlepony

oh crud made a mistake (sqrt(x^2+1)) - x(2x/2sqrt((x^(2)+1)))/(sqrt(x^2+1))^(2)

16. cuddlepony

17. ChrisV

because the final answer in the back of the book is $1/\sqrt{(x ^{2}+1)^{3}}$

18. ChrisV

thats the answer the book gives me

19. ChrisV

i know my first answer is correct but not simplified

20. cuddlepony

simplify

21. ChrisV

if i understood how to simplify it I wouldnt be here

22. ChrisV

lol

23. cuddlepony

fair enough :) i will help or try to my battery is about to die

24. cuddlepony

( (sqrt(x^2+1)) - x(2x/2sqrt((x^(2)+1))) )/(sqrt(x^2+1))^(2) = ( (sqrt(x^2+1)) - (x/sqrt(x^(2)+1)) )/ (sqrt(x^2+1))^(2) = (sqrt(x^2+1)) / (sqrt(x^2+1))^(2) - ( (x/sqrt(x^(2)+1)) / (sqrt(x^2+1))^(2) ) = 1 / (sqrt(x^2+1))^(2) - ( (x/sqrt(x^(2)+1)) / (sqrt(x^2+1))^(2) )

Truly, simplification and algebraic manipulation are the difficult parts of calculus; not the class' own namesake.

26. Mimi_x3

Lol, yes it is in the quotient rule, thats why i hate it. xD

27. ChrisV

somehow it simplifies to

28. cuddlepony

1 / (sqrt(x^2+1))^(2) - ( (x/sqrt(x^(2)+1)) / (sqrt(x^2+1))^(2) ) = 1 / (sqrt(x^2+1))^(2) - x(sqrt(x^2+1))^(2) )/sqrt(x^(2)+1) = 1 / (sqrt(x^2+1))^(2) - x(sqrt(x^2+1))/1

29. cuddlepony

1 / (sqrt(x^2+1)) - x(sqrt(x^2+1)) sorry made a mistake

30. ChrisV

$(x ^{2}+1)^{-3/2}(x ^{2}+1)/(x^2=1)$

31. ChrisV

/x^2+1) oops

32. cuddlepony

But yeah try multiplying both the top and the bottom by the conjugate: (x(sqrt(x^2+1))^(2) + 1) but as far as I would go to simplify this would be it it ( x(sqrt(x^2+1))^(2) - 1 )/(sqrt(x^2+1))

for final answer power in denominator should be 3/2

34. ChrisV

$(\sqrt{x ^{2}+1} - x^2/\sqrt{x^2+1})/(x^2+1)$

35. ChrisV

$(x^2+1-x^2)/\sqrt{x^2+1}/x^2+1$

36. ChrisV

$1/(x^2+1)(\sqrt{x^2+1})$

37. ChrisV

would $(x^2+1)(\sqrt{x^2+1})= \sqrt{(x^2+1)^3}$

38. ChrisV

because the final answer should be 1/sqrt(x+1)^3

40. ChrisV

i understand that marina same thing i have there

41. ChrisV

(x+1)^2/3 = sqrt((x+1)^2)

42. ChrisV

^3 oops

you're right y$(x^2+1) \sqrt{x^2+1}= (x^2+1)(x^2+1)^{1/2}= (x^2+1)^{3/2}$

44. ChrisV

i do believe $(x^2+1)(\sqrt{x^2+1}) = \sqrt{(x^2+1)^3}$

45. ChrisV

ok :) then the way i did that is right