1. **State the problem:** We need to find the derivative of the function $$f(u) = \csc^{-1}(8u + 1)$$. 2. **Recall the formula:** The derivative of $$y = \csc^{-1}(x)$$ is given by $$\frac{dy}{dx} = -\frac{1}{|x|\sqrt{x^2 - 1}}$$. 3. **Apply the chain rule:** Since our function is $$f(u) = \csc^{-1}(8u + 1)$$, let $$g(u) = 8u + 1$$. Then, $$f'(u) = \frac{d}{du} \csc^{-1}(g(u)) = \frac{d}{dg} \csc^{-1}(g(u)) \cdot g'(u)$$. 4. **Calculate the derivative of the inside function:** $$g'(u) = \frac{d}{du}(8u + 1) = 8$$. 5. **Substitute into the derivative formula:** $$f'(u) = -\frac{1}{|8u + 1| \sqrt{(8u + 1)^2 - 1}} \times 8$$. 6. **Simplify the expression:** $$f'(u) = -\frac{8}{|8u + 1| \sqrt{(8u + 1)^2 - 1}}$$. **Final answer:** $$f'(u) = -\frac{8}{|8u + 1| \sqrt{(8u + 1)^2 - 1}}$$.