Derivative of (2cos(t)+2sin(t))

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

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

      1. The derivative of cosine is negative sine:

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

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

   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:

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

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

   The result is: $- 2 \sin{\left(t \right)} + 2 \cos{\left(t \right)}$

2. Now simplify:

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

The answer is:

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