Derivative of cos(x+1)/2

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

   1. Let $u = x + 1$.

   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 $\frac{d}{d x} \left(x + 1\right)$:

      1. Differentiate $x + 1$ term by term:

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

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

         The result is: $1$

      The result of the chain rule is:

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

   So, the result is: $- \frac{\sin{\left(x + 1 \right)}}{2}$

2. Now simplify:

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

The answer is:

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