If `y=x^x` , find `(dy)/(dx)` at `x=e` .

Explanation

We have, `y=x^{x} ldots ldots ldots(i)`

Taking `log` on both sides,


`log y=log x^{x}`

`Rightarrow log y=x log x`

`Rightarrow frac{1}{y} frac{d y}{d x}=x frac{d}{d x}(log x)+log x frac{d}{d x}(x) `

`Rightarrow frac{1}{y} frac{d y}{d x}=x(frac{1}{x})+log x(1) `

`Rightarrow frac{1}{y} frac{d y}{d x}=1+log x `

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