Derivative of (sin^2)x+1

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

   1. Apply the product rule:

      $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

      $f{\left(x \right)} = \sin^{2}{\left(x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

      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:

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

      $g{\left(x \right)} = x$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

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

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

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

2. Now simplify:

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

The answer is:

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