1. **Problem:** Find the derivative of the function $$f(x) = x^{4 - x^2}$$ using logarithmic differentiation. 2. **Step 1: Apply logarithm to both sides** Take the natural logarithm: $$\ln f(x) = \ln \left(x^{4 - x^2}\right)$$ Using log rules: $$\ln f(x) = (4 - x^2) \ln x$$ 3. **Step 2: Differentiate both sides implicitly with respect to $$x$$** Recall $$\frac{d}{dx} \ln f(x) = \frac{f'(x)}{f(x)}$$ and use product rule on right side: $$\frac{f'(x)}{f(x)} = \frac{d}{dx} \left((4 - x^2) \ln x \right) = (4 - x^2) \frac{1}{x} + \ln x \frac{d}{dx} (4 - x^2)$$ 4. **Step 3: Compute the derivative of inside terms** $$\frac{d}{dx} (4 - x^2) = -2x$$ therefore: $$\frac{f'(x)}{f(x)} = \frac{4 - x^2}{x} + \ln x (-2x)$$ 5. **Step 4: Multiply both sides by $$f(x)$$ to solve for $$f'(x)$$** Recall $$f(x) = x^{4 - x^2}$$, so: $$f'(x) = x^{4 - x^2} \left( \frac{4 - x^2}{x} - 2x \ln x \right)$$ 6. **Final answer:** $$f'(x) = x^{4 - x^2} \left( \frac{4 - x^2}{x} - 2x \ln x \right)$$ This expresses the derivative clearly using logarithmic differentiation and product rule, showing all intermediate steps and explanations.