1. **State the problem:** Differentiate the function $$f(x) = \frac{\ln(x^4 + 6)}{x^5}$$ with respect to $$x$$. 2. **Recall the formula:** We will use the quotient rule for differentiation, which states: $$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$ where $$u = \ln(x^4 + 6)$$ and $$v = x^5$$. 3. **Find derivatives of $$u$$ and $$v$$:** - For $$u = \ln(x^4 + 6)$$, use the chain rule: $$u' = \frac{1}{x^4 + 6} \cdot \frac{d}{dx}(x^4 + 6) = \frac{1}{x^4 + 6} \cdot 4x^3 = \frac{4x^3}{x^4 + 6}$$ - For $$v = x^5$$: $$v' = 5x^4$$ 4. **Apply the quotient rule:** $$f'(x) = \frac{u'v - uv'}{v^2} = \frac{\frac{4x^3}{x^4 + 6} \cdot x^5 - \ln(x^4 + 6) \cdot 5x^4}{(x^5)^2}$$ 5. **Simplify numerator:** $$\frac{4x^3}{x^4 + 6} \cdot x^5 = \frac{4x^{8}}{x^4 + 6}$$ So numerator is: $$\frac{4x^{8}}{x^4 + 6} - 5x^4 \ln(x^4 + 6)$$ 6. **Write the full derivative:** $$f'(x) = \frac{\frac{4x^{8}}{x^4 + 6} - 5x^4 \ln(x^4 + 6)}{x^{10}} = \frac{4x^{8}}{(x^4 + 6) x^{10}} - \frac{5x^4 \ln(x^4 + 6)}{x^{10}}$$ 7. **Simplify powers of $$x$$:** $$\frac{4x^{8}}{(x^4 + 6) x^{10}} = \frac{4}{(x^4 + 6) x^{2}}$$ $$\frac{5x^4 \ln(x^4 + 6)}{x^{10}} = \frac{5 \ln(x^4 + 6)}{x^{6}}$$ 8. **Final answer:** $$f'(x) = \frac{4}{x^{2}(x^4 + 6)} - \frac{5 \ln(x^4 + 6)}{x^{6}}$$ This derivative shows how the function changes with respect to $$x$$, combining the logarithmic and polynomial parts carefully using the quotient and chain rules.