Derivative of sin^2x+2x

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

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

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

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

      1. The derivative of sine is cosine:

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

      The result of the chain rule is:

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

   4. 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: $2 \sin{\left(x \right)} \cos{\left(x \right)} + 2$

2. Now simplify:

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

The answer is:

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