1. **State the problem:** Simplify the expression $$\frac{x}{\sqrt{x^2 + y^2}} - \frac{1}{x^2 + y^2}$$. 2. **Identify the common denominator:** The terms have denominators $$\sqrt{x^2 + y^2}$$ and $$x^2 + y^2$$. Note that $$x^2 + y^2 = (\sqrt{x^2 + y^2})^2$$. 3. **Rewrite the first term with denominator $$x^2 + y^2$$:** Multiply numerator and denominator of the first term by $$\sqrt{x^2 + y^2}$$: $$\frac{x}{\sqrt{x^2 + y^2}} = \frac{x \cdot \sqrt{x^2 + y^2}}{(\sqrt{x^2 + y^2})^2} = \frac{x \sqrt{x^2 + y^2}}{x^2 + y^2}$$ 4. **Combine the terms over the common denominator:** $$\frac{x \sqrt{x^2 + y^2}}{x^2 + y^2} - \frac{1}{x^2 + y^2} = \frac{x \sqrt{x^2 + y^2} - 1}{x^2 + y^2}$$ 5. **Final simplified form:** $$\boxed{\frac{x \sqrt{x^2 + y^2} - 1}{x^2 + y^2}}$$ This is the simplest form without further assumptions about $$x$$ and $$y$$.