1. **State the problem:** Find the derivative of the function $$y = -8x^{-8} + 12\sqrt{x}$$ using the power rule. 2. **Recall the power rule:** For any function $$y = x^n$$, the derivative is $$\frac{dy}{dx} = nx^{n-1}$$. 3. **Rewrite the function:** Note that $$\sqrt{x} = x^{\frac{1}{2}}$$, so the function becomes $$y = -8x^{-8} + 12x^{\frac{1}{2}}$$. 4. **Differentiate each term separately:** - For $$-8x^{-8}$$, derivative is $$-8 \times (-8) x^{-8-1} = 64x^{-9}$$. - For $$12x^{\frac{1}{2}}$$, derivative is $$12 \times \frac{1}{2} x^{\frac{1}{2}-1} = 6x^{-\frac{1}{2}}$$. 5. **Combine the derivatives:** $$\frac{dy}{dx} = 64x^{-9} + 6x^{-\frac{1}{2}}$$. 6. **Final answer:** $$\boxed{\frac{dy}{dx} = 64x^{-9} + 6x^{-\frac{1}{2}}}$$