Derivative of a(cos*t+sin*t)

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

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

      1. The derivative of cosine is negative sine:

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

      2. The derivative of sine is cosine:

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

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

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

2. Now simplify:

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

The answer is:

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