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 am = 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,