Derivative of 2cos(x/2)

1. The derivative of a constant times a function is the constant times the derivative of the function.

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

   2. The derivative of cosine is negative sine:

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

   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:

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

   So, the result is: $- \sin{\left(\frac{x}{2} \right)}$

2. Now simplify:

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

The answer is:

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