1. **State the problem:** Find the derivative of the function $$y = x \cos^{-1} x - \sqrt{1 - x^2}$$. 2. **Recall formulas and rules:** - Derivative of $$\cos^{-1} x$$ is $$\frac{-1}{\sqrt{1 - x^2}}$$. - Use the product rule for $$x \cos^{-1} x$$: $$\frac{d}{dx}[uv] = u'v + uv'$$. - Derivative of $$\sqrt{1 - x^2} = (1 - x^2)^{1/2}$$ is $$\frac{1}{2}(1 - x^2)^{-1/2} \cdot (-2x) = \frac{-x}{\sqrt{1 - x^2}}$$. 3. **Apply product rule to $$x \cos^{-1} x$$:** - Let $$u = x$$, so $$u' = 1$$. - Let $$v = \cos^{-1} x$$, so $$v' = \frac{-1}{\sqrt{1 - x^2}}$$. 4. **Compute derivative of $$x \cos^{-1} x$$:** $$\frac{d}{dx}[x \cos^{-1} x] = 1 \cdot \cos^{-1} x + x \cdot \left(\frac{-1}{\sqrt{1 - x^2}}\right) = \cos^{-1} x - \frac{x}{\sqrt{1 - x^2}}$$. 5. **Compute derivative of $$- \sqrt{1 - x^2}$$:** $$\frac{d}{dx} \left(- \sqrt{1 - x^2}\right) = - \left(\frac{-x}{\sqrt{1 - x^2}}\right) = \frac{x}{\sqrt{1 - x^2}}$$. 6. **Combine derivatives:** $$y' = \cos^{-1} x - \frac{x}{\sqrt{1 - x^2}} + \frac{x}{\sqrt{1 - x^2}} = \cos^{-1} x$$. 7. **Final answer:** $$\boxed{\frac{dy}{dx} = \cos^{-1} x}$$. This means the derivative simplifies nicely to just $$\cos^{-1} x$$.