Derivative of x*sin(x)-x*cos(x)

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

   1. Apply the product rule:

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

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

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

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

      1. The derivative of sine is cosine:

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

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

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

      1. Apply the product rule:

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

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

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

         $g{\left(x \right)} = \cos{\left(x \right)}$; 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: $- x \sin{\left(x \right)} + \cos{\left(x \right)}$

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

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

2. Now simplify:

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

The answer is:

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