Differentiate the following function with respect to `x` : `(logx)^x+x^(logx)`

Explanation

Lety `=(log x)^{x}+x^{log x}`

`y=y_{1}+y_{2} ldots ldots ldots ldots ldots ldots ldots ldots` (2)

`frac{d y}{d x}=frac{d y_{1}}{d x}+frac{d y_{2}}{d x} ldots ldots ldots(3)`

Now take log of `y_{1}=(log x)^{x}`

`log y_{1}=x log (log x)`

`frac{1}{y_{2}} frac{d y_{2}}{d x}=(2 log x) times frac{1}{x}`

`frac{d y_{2}}{d x}=y_{2}(frac{2 log x}{x})=x^{log x}(frac{2 log x}{x}) ldots ldots ldots ldots ldots(5)`

Adding equation (4) and (5), we get:

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