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Partial Derivatives

  1. Jan 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Use the definition of partial deriviatives as limits to find fx(x,y) and fy(x,y).


    2. Relevant equations

    f(x,y) = [tex]\frac{x}{x + y^{2}}[/tex]


    3. The attempt at a solution

    I don't think this is right because I think I should have an answer of 1.

    fx(x,y) = lim h-> 0 [f(x+h,y) - f(x,y)]/h

    =lim h->0 [(x+h)/(x+h+y^2) - x/(x+y^2)]/h
    =lim h->0 [(x+h)/(x+h+y^2) - x/(x+y^2)]*1/h
    =lim h->0 (x+h)/(xh+h^2+(y^2)h) - x/(xh+(y^2)h)
    =lim h->0 ((x/h)+1)/(x+h+y^2) - (x/h)/(x+y^2)
    =1/(x+y^2)
     
  2. jcsd
  3. Jan 20, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I don't think the answer is 1. Look at your last line. You got things like (x/h). If you take lim h->0, that goes to infinity. You need to do enough algebra to cancel the h in the denominator before you can find a sensible limit. Combine (x+h)/(x+h+y^2) - x/(x+y^2) into single fraction and simplify the numerator before you take the limit.
     
  4. Jan 20, 2009 #3

    Mark44

    Staff: Mentor

    fx(x, y) is not equal to 1.

    I think you have an error in this line:
    =lim h->0 ((x/h)+1)/(x+h+y^2) - (x/h)/(x+y^2)

    I don't understand how you got to this expression from the one just before it.

    After taking the limit, you should end up with y^2/(x + y^2)^2
     
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