Finding the Derivative of (2/x)^(1/x)

[Nagihiko92]

Homework Statement

Find (2/x)^(1/x). Simplify your answer.

Homework Equations

The Attempt at a Solution

I let y = (2/x)^(1/x)
lny = ln(2/x)^(1/x)
lny= ln(2/x)/x
However, I get stuck here and don't know whether I should use e on both sides to get y by itself or to use implicit differentiation to get dy/dx directly.

 

[sylas]

I let y = (2/x)^(1/x)
lny = ln(2/x)^(1/x)
lny= ln(2/x)/x
However, I get stuck here and don't know whether I should use e on both sides to get y by itself or to use implicit differentiation to get dy/dx directly.

Minor typo... there's a missing bracket in the second line, but the third line is correct so that's okay.

If you use e on both sides you just get back what you started with. So that's no good. Try the other. Not sure what you mean by "implicit differentiation". Just differentiate both sides by x.

 

[Nagihiko92]

Ok so by using implicit differentiation and simplifying ln(2/x) as ln2 - lnx, I get
(1/y)(dy/dx) = ((x)(0-1/x) - (ln2-lnx)(1))/x
(1/y)(dy/dx) = (-x - ln2 - lnx)/x²
dy/dx = y(-x - ln2 - lnx)/x²

And my final answer turns out
dy/dx = (-xy - yln2 - ylnx)/x²

Did I do something wrong? It looks like I made it even more confusing.

 

[sylas]

Ok so by using implicit differentiation and simplifying ln(2/x) as ln2 - lnx, I get
(1/y)(dy/dx) = ((x)(0-1/x) - (ln2-lnx)(1))/x
(1/y)(dy/dx) = (-x - ln2 - lnx)/x²
dy/dx = y(-x - ln2 - lnx)/x²

And my final answer turns out
dy/dx = (-xy - yln2 - ylnx)/x²

Did I do something wrong? It looks like I made it even more confusing.

You can't have y in your answer. That was a variable you introduced. There is also an error in your differentiation of a quotient, and another error later on in the simplifications. But you are on the right track.

 

[Nagihiko92]

Ok, so I found my mistakes

(1/y)(dy/dx) = ((x)(-1/x) - (ln(2/x))(1))/x²
(1/y)(dy/dx) = (-1 - ln2 + lnx)/x²
dy/dx = y(-1 - ln2 + lnx)/x²

And if I can't leave y in my answer, I substituted y back in as

dy/dx = ((2/x)^(1/x)(-1 - ln2 + lnx))/x²
Should I factor in the (2/x)^(1/x) or leave it like that?

 

[sylas]

Looks good. Further changes are cosmetic and probably don't matter. Well done.