Derivative of (2x-1)^2*sin(x/2)

1. Apply the product rule:

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

   $f{\left(x \right)} = \left(2 x - 1\right)^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = 2 x - 1$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 1\right)$:

      1. Differentiate $2 x - 1$ term by term:

         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: $2$

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

         The result is: $2$

      The result of the chain rule is:

      $$8 x - 4$$

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

   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:

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

   The result is: $\frac{\left(2 x - 1\right)^{2} \cos{\left(\frac{x}{2} \right)}}{2} + \left(8 x - 4\right) \sin{\left(\frac{x}{2} \right)}$

2. Now simplify:

   $$\left(x - \frac{1}{2}\right) \left(\left(2 x - 1\right) \cos{\left(\frac{x}{2} \right)} + 8 \sin{\left(\frac{x}{2} \right)}\right)$$

The answer is:

$$\left(x - \frac{1}{2}\right) \left(\left(2 x - 1\right) \cos{\left(\frac{x}{2} \right)} + 8 \sin{\left(\frac{x}{2} \right)}\right)$$