Given: (log x)^x + x^{log x}

Let y= (log x)^x + x^{log x}

Let y = u + v

⇒ u = (log x)^x and v = x^{log x}

For, u =(log x)^x

Taking log on both sides, we get

log u =log (log x)^x

⇒log u = x.log (log(x))

Now, differentiate both sides with respect to x

For, v = x^{log x}

Taking log on both sides, we get

log v =log( x^{log x)}

⇒log v = log x. log x

Now, differentiate both sides with respect to x

Because, y = u + v