The discussion revolves around finding the derivative of the function (2/x)^(1/x) and simplifying the result. The subject area is calculus, specifically focusing on differentiation techniques and logarithmic properties.
The discussion is active, with participants identifying mistakes in their differentiation process and correcting them. There is a focus on ensuring that the final expression does not contain the variable y, and some guidance has been provided regarding the simplification of the derivative.
Participants mention a minor typo in the initial setup and discuss the implications of their differentiation steps, indicating a learning process around the rules of differentiation and logarithmic identities.
Minor typo... there's a missing bracket in the second line, but the third line is correct so that's okay.Nagihiko92 said: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.
Nagihiko92 said: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.