Derivative of 2(cosx+sinx)

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

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

      1. The derivative of cosine is negative sine:

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

      2. The derivative of sine is cosine:

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

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

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

2. Now simplify:

   $$2 \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}$$

The answer is:

$$2 \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}$$