Derivative of 2sin(x+1)-0.5

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

   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 sine is cosine:

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

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

         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:

         $$\cos{\left(x + 1 \right)}$$

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

   2. The derivative of the constant $- \frac{1}{2}$ is zero.

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

2. Now simplify:

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

The answer is:

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