Differentiate the following function with respect to x :

(log x)^x + x^{log x}.

Let y = (log x)^x + x^{log x} …(i)

Now, let p = (log)^x and q = x^{log x}

⇒ y = p + q …(ii)

Differentiating the above equation with respect to x, we get

…(A)

Now, consider p = (log x)^x

Firstly, taking log on both the sides, we get

log p = log[(log x)^x]

⇒ log p = x log(log x)

Differentiating with respect to x, we get

…(iii)

Now, consider q = x^{log x}

Taking log on both the sides, we get

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

⇒ log q = (log x)(log x)

⇒ log q = (log x)²

Differentiating with respect to x, we get