1. **State the problem:** Find the derivative of the function $$f(x) = \frac{1}{\sin(\ln x^2)}$$. 2. **Recall the formula:** To differentiate a function of the form $$f(x) = \frac{1}{g(x)}$$, use the quotient rule or rewrite as $$f(x) = (g(x))^{-1}$$ and apply the chain rule: $$f'(x) = - (g(x))^{-2} \cdot g'(x)$$. 3. **Identify inner function:** Here, $$g(x) = \sin(\ln x^2)$$. 4. **Differentiate inner function:** Use chain rule: - Derivative of $$\sin(u)$$ is $$\cos(u) \cdot u'$$. - Let $$u = \ln x^2$$. - Derivative of $$u$$ is $$\frac{d}{dx} \ln x^2 = \frac{1}{x^2} \cdot 2x = \frac{2}{x}$$. So, $$g'(x) = \cos(\ln x^2) \cdot \frac{2}{x}$$. 5. **Apply the derivative formula:** $$f'(x) = - \frac{g'(x)}{(g(x))^2} = - \frac{\cos(\ln x^2) \cdot \frac{2}{x}}{\sin^2(\ln x^2)} = - \frac{2 \cos(\ln x^2)}{x \sin^2(\ln x^2)}$$. 6. **Final answer:** $$\boxed{f'(x) = - \frac{2 \cos(\ln x^2)}{x \sin^2(\ln x^2)}}$$. This derivative tells us the rate of change of the function $$f(x)$$ with respect to $$x$$, using the chain rule and quotient rule concepts.