1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Find the second derivative

  1. Jul 21, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the second derivative of 3(x^2)y+y+x=x^5


    2. Relevant equations
    Find the first derivative using implicit differentiation.
    Find the second by using quotient rule.


    3. The attempt at a solution
    So I found the first derivative to be

    dy/dx = 5(x^4)-6xy-1 / 3(x^2)+1

    I get very lost after this trying to find the second derivative using the quotient rule as there is the y variable still in the equation. Any help please?
     
  2. jcsd
  3. Jul 21, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    The equation for y' is wrong without parentheses.
    Isolate y from the first equation 3(x^2)y+y+x=x^5 and sub into y'.

    ehild
     
  4. Jul 21, 2012 #3

    rock.freak667

    User Avatar
    Homework Helper

    Put u = 5(x^4)-6xy-1, then get du/dx using implicit differentiation and put v= 3(x^2)+1 and get dv/dx.

    Then just put it into your formula for the quotient rule. But in du/dx you will have a term with dy/dx in it (which you know from the first derivative).
     
  5. Jul 21, 2012 #4
    Finding du/dx I get 20(x^3)-6x(dy/dx)-6y and dv/dx I get 6x. Is that correct?

    This is what I tried earlier but got very lost trying to sub in dy/dx
     
    Last edited: Jul 21, 2012
  6. Jul 21, 2012 #5

    ehild

    User Avatar
    Homework Helper
    Gold Member

    It is correct now.:smile:

    Show your further work.

    ehild
     
  7. Jul 21, 2012 #6
    so then using the quotient rule

    dy/dx = (v(du/dx)-u(dv/dx))/(v^2)

    = ((3(x^2)+1)*(20(x^3)-(6x((5(x^4)-6xy-1)) / (3(x^2)+1))-6y) - (5(x^4)-6xy-1)* (6x)) / ((3(x^2)+1)^2)

    sorry it looks very confusing like that if I knew how to make an image i would.

    My calculator gives a result of (2(10(x^3)+27(x^2)(y)+6x-3y) / (3(x^2)+1)^2)
    and I have not been able to get close to that answer.
     
  8. Jul 21, 2012 #7
    factoring the above I get a result of

    (-60(x^5)+20(x^3)+72(x^2)(y)+12x-6y) / ((3x^2+1)^2)

    or

    (2(-30(x^5)+10(x^3)+36(x^2)(y)+6x-3y)) / ((3x^2+1)^2)
     
  9. Jul 21, 2012 #8

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Finding y' by implicit diferentiation is a good idea and I see no reason not to do the same for the second derivative.

    [itex]3x^2y+y+x=x^5[/itex] so
    [itex]6xy+ 3x^2y'+ y'+ 1= 5x^4[/itex]

    Differentiating, implicitely, again,
    [itex]6y+ 6xy'+ 6xy'+ 3x^2y''+ y''= 20x^3[/itex]
    [itex]6y+ 12xy'+ (3x^2+ 1)y''= 20x^3[/itex]

    You can solve
    [itex]3x^2y+y+x=x^5[/itex]
    for [itex]y= \frac{x^5- x}{3x^2+ 1}[/itex] and
    [itex]6xy+ 3x^2y'+ y'+ 1= 5x^4[/itex]
    for [itex]y'= \frac{5x^4- 6xy+ 1}{3x^2+ 1}[/itex]
    and put those into [itex]y''= \frac{20x^3- 12xy'- 6y}{3x^2+ 1}[/itex]
    if you like but my experience is that you seldom have to solve for y'' in terms of x only.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Find the second derivative
  1. Find second derivative (Replies: 3)

Loading...