Derivative of cos(ax)+sin(ax)

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

   1. Let $u = a x$.

   2. The derivative of cosine is negative sine:

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

   3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} a x$:

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

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $a$

      The result of the chain rule is:

      $$- a \sin{\left(a x \right)}$$

   4. Let $u = a x$.

   5. The derivative of sine is cosine:

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

   6. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} a x$:

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

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $a$

      The result of the chain rule is:

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

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

2. Now simplify:

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

The answer is:

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