1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{1}{\sqrt{x^6 + 2x + 1}}.$$\n\n2. **Rewrite the function:** To differentiate easily, rewrite the function using exponents: $$f(x) = (x^6 + 2x + 1)^{-\frac{1}{2}}.$$\n\n3. **Recall the chain rule:** If $$f(x) = g(h(x))$$ then $$f'(x) = g'(h(x)) \cdot h'(x).$$ Here, $$g(u) = u^{-\frac{1}{2}}$$ and $$h(x) = x^6 + 2x + 1.$$\n\n4. **Differentiate the outer function:** Using the power rule, $$g'(u) = -\frac{1}{2} u^{-\frac{3}{2}}.$$\n\n5. **Differentiate the inner function:** $$h'(x) = 6x^5 + 2.$$\n\n6. **Apply the chain rule:** $$f'(x) = g'(h(x)) \cdot h'(x) = -\frac{1}{2} (x^6 + 2x + 1)^{-\frac{3}{2}} \cdot (6x^5 + 2).$$\n\n7. **Write the final answer:** $$\boxed{f'(x) = -\frac{6x^5 + 2}{2 (x^6 + 2x + 1)^{\frac{3}{2}}}}.$$\n\nThis derivative tells us how the function changes at any point $$x$$ by combining the rate of change of the inside function and the outside function using the chain rule.