1. **Problem:** Find the derivative of the function $$f(x) = \frac{1}{(1 - x^2)^{3/2}}$$. 2. **Formula and rules:** Use the chain rule and power rule. For $$f(x) = (g(x))^n$$, $$f'(x) = n(g(x))^{n-1} g'(x)$$. 3. **Rewrite:** $$f(x) = (1 - x^2)^{-3/2}$$. 4. **Differentiate:** $$f'(x) = -\frac{3}{2} (1 - x^2)^{-5/2} \cdot (-2x)$$ 5. **Simplify:** $$f'(x) = -\frac{3}{2} \cdot (-2x) (1 - x^2)^{-5/2} = 3x (1 - x^2)^{-5/2}$$ 6. **Final answer:** $$\boxed{f'(x) = \frac{3x}{(1 - x^2)^{5/2}}}$$