1. **State the problem:** Differentiate the expression $$10r^{5} + 7r^{3} - 11r + 8$$ with respect to $$r$$. 2. **Recall the power rule for differentiation:** For any term $$ar^n$$, the derivative with respect to $$r$$ is $$a n r^{n-1}$$. 3. **Apply the power rule to each term:** - Derivative of $$10r^{5}$$ is $$10 \times 5 r^{5-1} = 50r^{4}$$. - Derivative of $$7r^{3}$$ is $$7 \times 3 r^{3-1} = 21r^{2}$$. - Derivative of $$-11r$$ is $$-11 \times 1 r^{1-1} = -11$$. - Derivative of constant $$8$$ is $$0$$. 4. **Combine all derivatives:** $$\frac{d}{dr} \left(10r^{5} + 7r^{3} - 11r + 8\right) = 50r^{4} + 21r^{2} - 11$$. 5. **Final answer:** $$\boxed{50r^{4} + 21r^{2} - 11}$$