Derivative of y=x(lnx-1):x*2

1. The derivative of a constant times a function is the constant times the derivative of the function.

   1. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = x \left(\log{\left(x \right)} - 1\right)$ and $g{\left(x \right)} = x$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

      1. Apply the product rule:

         $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

         $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

         1. Apply the power rule: $x$ goes to $1$

         $g{\left(x \right)} = \log{\left(x \right)} - 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

         1. Differentiate $\log{\left(x \right)} - 1$ term by term:

            1. The derivative of the constant $-1$ is zero.

            2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$.

            The result is: $\frac{1}{x}$

         The result is: $\log{\left(x \right)}$

      To find $\frac{d}{d x} g{\left(x \right)}$:

      1. Apply the power rule: $x$ goes to $1$

      Now plug in to the quotient rule:

      $\frac{- x \left(\log{\left(x \right)} - 1\right) + x \log{\left(x \right)}}{x^{2}}$

   So, the result is: $\frac{2 \left(- x \left(\log{\left(x \right)} - 1\right) + x \log{\left(x \right)}\right)}{x^{2}}$

2. Now simplify:

   $$\frac{2}{x}$$

The answer is:

$$\frac{2}{x}$$