Derivative of cosx^2+x^2

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

   1. Let $u = \cos{\left(x \right)}$.

   2. Apply the power rule: $u^{2}$ goes to $2 u$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$:

      1. The derivative of cosine is negative sine:

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

      The result of the chain rule is:

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

   4. Apply the power rule: $x^{2}$ goes to $2 x$

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

2. Now simplify:

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

The answer is: