1. **State the problem:** Differentiate the function $$f(x) = \frac{(3 \cdot \sqrt[3]{x} + 2)^2}{2x^5}.$$ 2. **Rewrite the function for clarity:** Express the cube root as a power: $$\sqrt[3]{x} = x^{\frac{1}{3}}.$$ So, $$f(x) = \frac{(3x^{\frac{1}{3}} + 2)^2}{2x^5} = \frac{g(x)}{h(x)},$$ where $$g(x) = (3x^{\frac{1}{3}} + 2)^2$$ and $$h(x) = 2x^5.$$ 3. **Use the quotient rule:** If $$f(x) = \frac{g(x)}{h(x)},$$ then $$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}.$$ 4. **Find \(g'(x)\):** First, expand or use chain rule: $$g(x) = (3x^{\frac{1}{3}} + 2)^2,$$ let $$u = 3x^{\frac{1}{3}} + 2,$$ then $$g'(x) = 2u \cdot u' = 2(3x^{\frac{1}{3}} + 2)(3 \cdot \frac{1}{3}x^{-\frac{2}{3}}) = 2(3x^{\frac{1}{3}} + 2)(x^{-\frac{2}{3}}).$$ 5. **Find \(h'(x)\):** $$h(x) = 2x^5 \implies h'(x) = 2 \cdot 5x^{4} = 10x^{4}.$$ 6. **Apply the quotient rule:** $$f'(x) = \frac{2(3x^{\frac{1}{3}} + 2)(x^{-\frac{2}{3}})(2x^5) - (3x^{\frac{1}{3}} + 2)^2(10x^4)}{(2x^5)^2}.$$ 7. **Simplify numerator step-by-step:** - First term: $$2(3x^{\frac{1}{3}} + 2)(x^{-\frac{2}{3}})(2x^5) = 4(3x^{\frac{1}{3}} + 2)x^{5 - \frac{2}{3}} = 4(3x^{\frac{1}{3}} + 2)x^{\frac{15}{3} - \frac{2}{3}} = 4(3x^{\frac{1}{3}} + 2)x^{\frac{13}{3}}.$$ - Second term: $$ (3x^{\frac{1}{3}} + 2)^2 (10x^4).$$ So numerator becomes: $$4(3x^{\frac{1}{3}} + 2)x^{\frac{13}{3}} - 10(3x^{\frac{1}{3}} + 2)^2 x^4.$$ 8. **Simplify denominator:** $$(2x^5)^2 = 4x^{10}.$$ 9. **Final expression for derivative:** $$f'(x) = \frac{4(3x^{\frac{1}{3}} + 2)x^{\frac{13}{3}} - 10(3x^{\frac{1}{3}} + 2)^2 x^4}{4x^{10}}.$$ 10. **Optional further simplification:** Divide numerator and denominator by 2 or factor common terms if needed, but this form shows clear intermediate steps and derivative. **Answer:** $$\boxed{f'(x) = \frac{4(3x^{\frac{1}{3}} + 2)x^{\frac{13}{3}} - 10(3x^{\frac{1}{3}} + 2)^2 x^4}{4x^{10}}}.$$