Derivative of 1+sin^2x

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

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

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

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

   4. 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:

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

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

2. Now simplify:

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

The answer is:

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