Derivative of sin(x)*cos(x)+x

1. Differentiate $x + \sin{\left(x \right)} \cos{\left(x \right)}$ term by term:

   1. Apply the product rule:

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

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

      1. The derivative of sine is cosine:

         $$\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$$

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

      1. The derivative of cosine is negative sine:

         $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

      The result is: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$

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

   The result is: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + 1$

2. Now simplify:

   $$2 \cos^{2}{\left(x \right)}$$

The answer is:

$$2 \cos^{2}{\left(x \right)}$$