Is the Derivative of y=x^(1/x) Correct?

[dylanhouse]

Homework Statement

Find the derivative of y=x^1/x

Homework Equations

Chain rule and logarithmic differentiation.

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 :$

 

[stephenkeiths]

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?

 

[dylanhouse]

Should it read:

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

 

[stephenkeiths]

close shouldn't the derivative of x^-1 be -1*x^-2??

 

[dylanhouse]

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)

 

[stephenkeiths]

Looks good to me. Just plug in y like you did in the last step of the 1st post and you're golden.

 

[dylanhouse]

Thanks a bunch :)

 

[skiller]

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.