Derivative of x^2(sin(0.5x)+1)

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = \sin{\left(\frac{x}{2} \right)} + 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

      1. Let $u = \frac{x}{2}$.

      2. The derivative of sine is cosine:

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

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

         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: $\frac{1}{2}$

         The result of the chain rule is:

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

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

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

   The result is: $\frac{x^{2} \cos{\left(\frac{x}{2} \right)}}{2} + 2 x \left(\sin{\left(\frac{x}{2} \right)} + 1\right)$

2. Now simplify:

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

The answer is:

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