Differentiate the following function with respect to `x:x^(1/x)`

Explanation

Let `y = x^(1/x) `

Taking log both the sides:

{log xa = a log x}

Differentiating with respect to x:

`(d(logy))/(dx)=(d(1/x logx))/(dx)`

`(d(logy))/(dx)=1/x times (d(logx))/(dx)+log x times (d(x^-1))/(dx)`

Using products rule, `(d(uv))/(dx)=u(dv)/(dx)+v(du)/(dx)`
`Rightarrow 1/y (dy)/(dx)=1/x times (1dx)/(xdx)+log x(-1x^2)`

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