1. **State the problem:** Find the first derivative of the function $$y = \arcsin\left(\frac{a}{x}\right)$$ where $$x > 2$$. 2. **Recall the formula:** The derivative of $$y = \arcsin(u)$$ with respect to $$x$$ is $$\frac{dy}{dx} = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}$$. 3. **Identify $$u$$:** Here, $$u = \frac{a}{x}$$. 4. **Compute $$\frac{du}{dx}$$:** $$\frac{du}{dx} = \frac{d}{dx} \left(\frac{a}{x}\right) = -\frac{a}{x^2}$$. 5. **Substitute into the derivative formula:** $$\frac{dy}{dx} = \frac{1}{\sqrt{1-\left(\frac{a}{x}\right)^2}} \cdot \left(-\frac{a}{x^2}\right)$$. 6. **Simplify the expression inside the square root:** $$1 - \left(\frac{a}{x}\right)^2 = 1 - \frac{a^2}{x^2} = \frac{x^2 - a^2}{x^2}$$. 7. **Rewrite the derivative:** $$\frac{dy}{dx} = -\frac{a}{x^2} \cdot \frac{1}{\sqrt{\frac{x^2 - a^2}{x^2}}} = -\frac{a}{x^2} \cdot \frac{1}{\frac{\sqrt{x^2 - a^2}}{x}} = -\frac{a}{x^2} \cdot \frac{x}{\sqrt{x^2 - a^2}}$$. 8. **Simplify further:** $$\frac{dy}{dx} = -\frac{a}{x \sqrt{x^2 - a^2}}$$. **Final answer:** $$\boxed{\frac{dy}{dx} = -\frac{a}{x \sqrt{x^2 - a^2}}}$$