1. The problem is to find the derivative of the function $$f(x) = (3x + 2)^2 + (3x + 2)^4$$. 2. We use the chain rule for differentiation: if $$f(x) = g(h(x))$$, then $$f'(x) = g'(h(x)) \cdot h'(x)$$. 3. Let $$u = 3x + 2$$. Then the function becomes $$f(x) = u^2 + u^4$$. 4. Differentiate each term with respect to $$x$$: $$\frac{d}{dx} u^2 = 2u \cdot \frac{du}{dx}$$ $$\frac{d}{dx} u^4 = 4u^3 \cdot \frac{du}{dx}$$ 5. Since $$u = 3x + 2$$, we have $$\frac{du}{dx} = 3$$. 6. Substitute back: $$f'(x) = 2u \cdot 3 + 4u^3 \cdot 3 = 6u + 12u^3$$ 7. Replace $$u$$ with $$3x + 2$$: $$f'(x) = 6(3x + 2) + 12(3x + 2)^3$$ This is the derivative of the function.