Differentiate (log x)^x with respect to log x.

Let u = (log x)^x and v = log x.

We need to differentiate u with respect to v that is find.

We have u = (log x)^x

Taking log on both sides, we get

log u = log(log x)^x

⇒ log u = x × log(log x) [∵ log a^m = m × log a]

On differentiating both the sides with respect to x, we get

Recall that (uv)’ = vu’ + uv’ (product rule)

We know and

But, u = (log x)^x and

Now, on differentiating v with respect to x, we get

We have

Thus,