1. **State the problem:** Differentiate the function $y = (2x+3)^8$ with respect to $x$. 2. **Formula used:** We use the chain rule for differentiation, which states: $$\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$$ 3. **Identify inner and outer functions:** - Outer function: $f(u) = u^8$ - Inner function: $g(x) = 2x + 3$ 4. **Differentiate outer function:** $$f'(u) = 8u^{7}$$ 5. **Differentiate inner function:** $$g'(x) = 2$$ 6. **Apply chain rule:** $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x) = 8(2x+3)^7 \cdot 2$$ 7. **Simplify:** $$\frac{dy}{dx} = 16(2x+3)^7$$ **Final answer:** $$\boxed{\frac{dy}{dx} = 16(2x+3)^7}$$