Find `(dy)/(dx),` `y=e^x+10^x+x^x`

Explanation

We have, `y=e^{x}+10^{x}+x^{x}`

`&Rightarrow y=e^{x}+10^{x}+e^{log x^{x}} `

`&Rightarrow y=e^{x}+10^{x}+e^{x log x}`

`&frac{d y}{d x}=frac{d}{d x}(e^{x})+frac{d}{d x}(10^{x})+frac{d}{d x}(e^{x log x}) `

`&=e^{x}+10^{x} log 10+e^{x log x} frac{d}{d x}(x log x) `

`&=e^{x}+10^{x} log 10+e^{x log x}[x frac{d}{d x}(log x)+log x frac{d}{d x}(x)] `

`&=e^{x}+10^{x} log 10+e^{log x^{x}}[x(frac{1}{x})+log x] `

`&=e^{x}+10^{x} log 10+x^{x}[1+log x] `

...