1. **State the problem:** Find the derivative of the function $$y = (x^2 + 3)^4 (2x^3 - 5)^3$$. 2. **Formula used:** We will use the product rule and the chain rule. - Product rule: $$\frac{d}{dx}[u \cdot v] = u'v + uv'$$ - Chain rule: $$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$ 3. **Identify parts:** Let $$u = (x^2 + 3)^4$$ and $$v = (2x^3 - 5)^3$$. 4. **Find $$u'$$:** - Using chain rule: $$u' = 4(x^2 + 3)^3 \cdot \frac{d}{dx}(x^2 + 3) = 4(x^2 + 3)^3 \cdot 2x = 8x(x^2 + 3)^3$$. 5. **Find $$v'$$:** - Using chain rule: $$v' = 3(2x^3 - 5)^2 \cdot \frac{d}{dx}(2x^3 - 5) = 3(2x^3 - 5)^2 \cdot 6x^2 = 18x^2(2x^3 - 5)^2$$. 6. **Apply product rule:** $$\frac{dy}{dx} = u'v + uv' = 8x(x^2 + 3)^3 (2x^3 - 5)^3 + (x^2 + 3)^4 18x^2 (2x^3 - 5)^2$$. 7. **Factor common terms:** - Common factors: $$x (x^2 + 3)^3 (2x^3 - 5)^2$$ - So, $$\frac{dy}{dx} = x (x^2 + 3)^3 (2x^3 - 5)^2 \left[8 (2x^3 - 5) + 18x (x^2 + 3) \right]$$. 8. **Simplify inside the bracket:** - $$8 (2x^3 - 5) + 18x (x^2 + 3) = 16x^3 - 40 + 18x^3 + 54x = (16x^3 + 18x^3) + 54x - 40 = 34x^3 + 54x - 40$$. 9. **Final derivative:** $$\frac{dy}{dx} = x (x^2 + 3)^3 (2x^3 - 5)^2 (34x^3 + 54x - 40)$$. This is the derivative of the given function.