Derivative of x+cos2x

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

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

   2. Let $u = 2 x$.

   3. The derivative of cosine is negative sine:

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

   4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

      1. 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 of the chain rule is:

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

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

The answer is:

$$1 - 2 \sin{\left(2 x \right)}$$