What is the inverse of y= x ln(x)?

As far as finding the inverse go that looks right. If you plug the original function into its inverse and get X. What you found as an inverse is correct e^(x/xln(x)) clearly give x because in x/xln(x) the Xs cancel and you're left with e^ln(x) which is just x. As far as the domain of the inverse goes, it looks like it's from positive to negative infinity. As, you can plug in any x value(domain only refers to x values) and get back an answer.

To the first answer, he isn't finding the rate of change of anything so I hardly see as to how derivatives are related to this.

Obviously, getting x on its own is gonna be problematic. We don't even have a dy/dx term so we can't solve a differential equation on it.

However, we know that the range of one function is the domain of it's inverse.

let's assume g = ln(x). What is the range of g? That would affect the range of y. The range of g is:

-inf<=g<=inf or g goes between negative and positive infinity.

Hence xln(x) would go from negative infinity to positive infinity.

So your domain of the inverse would be such that x goes from negative infinity to positive infinity.

Hope this helps! ^^