Derivative of (С1+x)lnx-2x+С2

1. Differentiate $c_{2} + \left(- 2 x + \left(c_{1} + x\right) \log{\left(x \right)}\right)$ term by term:

   1. Differentiate $- 2 x + \left(c_{1} + x\right) \log{\left(x \right)}$ term by term:

      1. Apply the product rule:

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

         $f{\left(x \right)} = c_{1} + x$; to find $\frac{d}{d x} f{\left(x \right)}$:

         1. Differentiate $c_{1} + x$ term by term:

            1. The derivative of the constant $c_{1}$ is zero.

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

            The result is: $1$

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

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

         The result is: $\log{\left(x \right)} + \frac{c_{1} + x}{x}$

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

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

         So, the result is: $-2$

      The result is: $\log{\left(x \right)} - 2 + \frac{c_{1} + x}{x}$

   2. The derivative of the constant $c_{2}$ is zero.

   The result is: $\log{\left(x \right)} - 2 + \frac{c_{1} + x}{x}$

2. Now simplify:

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

The answer is:

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