Derivative of e^xcosx

1. Apply the product rule:

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

   $f{\left(x \right)} = e^{x}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. The derivative of $e^{x}$ is itself.

   $g{\left(x \right)} = \cos{\left(x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. The derivative of cosine is negative sine:

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

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

2. Now simplify:

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

The answer is:

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