1. **State the problem:** Find the derivative of the function $$f(x) = 5x^8 + 5x^3 - 6x - 6$$. 2. **Recall the derivative rules:** - The derivative of $$x^n$$ is $$nx^{n-1}$$. - The derivative of a constant is 0. - The derivative of a sum/difference is the sum/difference of the derivatives. 3. **Apply the power rule to each term:** - Derivative of $$5x^8$$ is $$5 \times 8x^{8-1} = 40x^7$$. - Derivative of $$5x^3$$ is $$5 \times 3x^{3-1} = 15x^2$$. - Derivative of $$-6x$$ is $$-6 \times 1x^{1-1} = -6$$. - Derivative of $$-6$$ is 0. 4. **Combine all derivatives:** $$\frac{d}{dx}(5x^8 + 5x^3 - 6x - 6) = 40x^7 + 15x^2 - 6$$. 5. **Final answer:** $$\boxed{40x^7 + 15x^2 - 6}$$