Derivative of x(lnx+1)

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 $\log{\left(x \right)}$ is $\frac{1}{x}$.

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

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

   The result is: $\log{\left(x \right)} + 2$

The answer is:

$$\log{\left(x \right)} + 2$$