1. **Statement of the problem:** Find the derivative of the function: $$f(x) = \left(\sqrt[3]{4x^{2} - 5}\right)^{-1} = \frac{1}{(4x^{2} - 5)^{1/3}}$$ 2. **Rewrite the function:** Using exponent notation, $$f(x) = (4x^{2} - 5)^{-1/3}$$ 3. **Apply the chain rule:** The derivative of $f(x) = u^{n}$ where $u = 4x^{2} - 5$ and $n = -\frac{1}{3}$ is $$f'(x) = n u^{n-1} \cdot u'$$ 4. **Calculate $u'$:** $$u' = \frac{d}{dx}(4x^{2} - 5) = 8x$$ 5. **Substitute values into the derivative formula:** $$f'(x) = -\frac{1}{3} (4x^{2} - 5)^{-\frac{4}{3}} \cdot 8x$$ 6. **Simplify the expression:** $$f'(x) = -\frac{8x}{3} (4x^{2} - 5)^{-\frac{4}{3}}$$ This is the correct derivative. 7. **Check the incorrect derivative given:** The provided derivative was $$f'(x) = - \frac{1}{6x^{3}} \cdot (4x^{2} - 5)^{-\frac{2}{3}}$$ This is incorrect because the derivative does not have $x^{3}$ in the denominator and the exponent of $(4x^{2} - 5)$ is wrong. **Final answer:** $$\boxed{f'(x) = -\frac{8x}{3} (4x^{2} - 5)^{-\frac{4}{3}}}$$ This completes the differentiation process with detailed steps.