1. **State the problem:** Differentiate the function $$f(x) = \frac{x^2 \sin(x) - 23 \log(x^2)}{\sqrt{x}}$$ with respect to $$x$$. 2. **Rewrite the function:** To simplify differentiation, express $$f(x)$$ as $$f(x) = (x^2 \sin(x) - 23 \log(x^2)) \cdot x^{-\frac{1}{2}}$$. 3. **Recall differentiation rules:** - Product rule: $$(uv)' = u'v + uv'$$ - Chain rule for logarithm: $$\frac{d}{dx} \log(g(x)) = \frac{g'(x)}{g(x)}$$ - Derivative of $$\sin(x)$$ is $$\cos(x)$$ - Derivative of $$x^n$$ is $$nx^{n-1}$$ 4. **Define:** $$u = x^2 \sin(x) - 23 \log(x^2)$$ $$v = x^{-\frac{1}{2}}$$ 5. **Differentiate $$u$$:** - Derivative of $$x^2 \sin(x)$$ using product rule: $$\frac{d}{dx}(x^2 \sin(x)) = 2x \sin(x) + x^2 \cos(x)$$ - Derivative of $$-23 \log(x^2)$$: Since $$\log(x^2) = 2 \log(x)$$, $$\frac{d}{dx}(-23 \log(x^2)) = -23 \cdot \frac{d}{dx}(2 \log(x)) = -46 \cdot \frac{1}{x} = -\frac{46}{x}$$ So, $$u' = 2x \sin(x) + x^2 \cos(x) - \frac{46}{x}$$ 6. **Differentiate $$v$$:** $$v = x^{-\frac{1}{2}}$$ $$v' = -\frac{1}{2} x^{-\frac{3}{2}}$$ 7. **Apply product rule:** $$f'(x) = u'v + uv'$$ $$= \left(2x \sin(x) + x^2 \cos(x) - \frac{46}{x}\right) x^{-\frac{1}{2}} + \left(x^2 \sin(x) - 23 \log(x^2)\right) \left(-\frac{1}{2} x^{-\frac{3}{2}}\right)$$ 8. **Simplify terms:** - First term: $$\left(2x \sin(x) + x^2 \cos(x) - \frac{46}{x}\right) x^{-\frac{1}{2}} = 2x^{\frac{1}{2}} \sin(x) + x^{\frac{3}{2}} \cos(x) - 46 x^{-\frac{3}{2}}$$ - Second term: $$-\frac{1}{2} x^{-\frac{3}{2}} (x^2 \sin(x) - 23 \log(x^2)) = -\frac{1}{2} x^{\frac{1}{2}} \sin(x) + \frac{23}{2} x^{-\frac{3}{2}} \log(x^2)$$ 9. **Combine like terms:** $$f'(x) = \left(2x^{\frac{1}{2}} \sin(x) - \frac{1}{2} x^{\frac{1}{2}} \sin(x)\right) + x^{\frac{3}{2}} \cos(x) - 46 x^{-\frac{3}{2}} + \frac{23}{2} x^{-\frac{3}{2}} \log(x^2)$$ $$= \frac{3}{2} x^{\frac{1}{2}} \sin(x) + x^{\frac{3}{2}} \cos(x) - 46 x^{-\frac{3}{2}} + \frac{23}{2} x^{-\frac{3}{2}} \log(x^2)$$ **Final answer:** $$f'(x) = \frac{3}{2} x^{\frac{1}{2}} \sin(x) + x^{\frac{3}{2}} \cos(x) - 46 x^{-\frac{3}{2}} + \frac{23}{2} x^{-\frac{3}{2}} \log(x^2)$$