Differentiate the function.
$ f(x) = x \ln x - x $
this problem. We're going to be taking the derivative mostly with the main portion reason why we're doing this because we're doing it with the natural log. Um, we are computing derivatives of logarithmic functions for the section. So what we have is the ffx being equal to ex kind of natural log of acts minus X. So what we want to do is take the derivative this So we want f prime of X, and we see that this is going to be equal Thio. First, we're gonna have a chain rule, so we'll have X times, the derivative of natural Log of X, which we know is just going to give us one. And then that's going to be one plus the natural log of acts times the derivative X, which is just one. And then lastly, we'll have a minus one here. So we do that, we just get rid of this and it results in. I'm just having the natural log of X as our answer eso we see that will be our final answer for the derivative. And in fact, we see that this is the same thing as if we work a graph f prime of X. It's the exact same graph we look here s so that just shows us that we did it properly in taking the derivative.