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Is this derivative correct?

  1. Oct 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the derivative of y=x1/x

    2. Relevant equations

    Chain rule and logarithmic differentiation.

    3. The attempt at a solution

    y=x^(1/x)
    lny=lnx^(1/x)
    lny=(1/x)lnx
    (1/y)y'=(1/x)lnx
    y'=y((1/x)lnx)
    y'=y((1/x)(1/x)+(lnx)(1/x))
    y'=y((1/x^2)+(lnx/x))
    y'=x^(1/x)((1/x^2)+(lnx/x))

    Sorry for all the brackets, and if they are not correct :$
     
  2. jcsd
  3. Oct 21, 2012 #2
    In your 6th line I think I see a mistake you're using the product rule it seems? In the second part what did you take the derivative of?
     
  4. Oct 21, 2012 #3
    Should it read:

    y'=y((1/x)(1/x)+lnx(1/x^2))?
     
  5. Oct 21, 2012 #4
    close shouldn't the derivative of x^-1 be -1*x^-2??
     
  6. Oct 21, 2012 #5
    So,
    y'=y((1/x)(1/x)+lnx(-1/x^2))
    y'=y((1/x^2)-(lnx/x^2))
    y'=y(1-lnx/x^2)
     
  7. Oct 21, 2012 #6
    Looks good to me. Just plug in y like you did in the last step of the 1st post and you're golden.
     
  8. Oct 21, 2012 #7
    Thanks a bunch :)
     
  9. Oct 22, 2012 #8
    You have a problem with parentheses, but more importantly, lines 4 and 5 are very wrong.

    You cannot differentiate the LHS and still equate it with the RHS without differentiating the RHS also.

    However you still got there - as long as you correct your parentheses.
     
    Last edited: Oct 22, 2012
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