Derivative of (sin(x))*(cos(x))-2*x+1

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

   1. Differentiate $- 2 x + \sin{\left(x \right)} \cos{\left(x \right)}$ 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. 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: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - 2$

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

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

2. Now simplify:

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

The answer is:

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