Find `(dy)/(dx),` `y=x^x+x^(1/x)`

Explanation

here

`y=x^{x}+x^{1 / x}`

`=e^{log x^{x}+e^{log x^{frac{1}{x}}}}`

`y=e^{x log x}+e^{(frac{1}{x} log x)}`

`{[.` Since `.log a^{b}=b log a] }`

Differentiating it with respect tox using the chain rule and product rule,

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

`=e^{x log x}+frac{d}{d x}(x log x)+e^{frac{1}{x} log x} frac{d}{d x}(frac{1}{x} log x)`

...