Derivative of 2(1+sinx)½+10

1. Differentiate $2 \left(\sin{\left(x \right)} + 1\right) \frac{1}{2} + 10$ term by term:

   1. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = 2 \sin{\left(x \right)} + 2$ and $g{\left(x \right)} = 2$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

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

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

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

            1. The derivative of sine is cosine:

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

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

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

      To find $\frac{d}{d x} g{\left(x \right)}$:

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

      Now plug in to the quotient rule:

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

   2. The derivative of the constant $10$ is zero.

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

The answer is:

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