1. **Problem:** Differentiate $y = (3x+1)^5$ with respect to $x$. 2. **Formula:** Use the chain rule for differentiation: If $y = [u(x)]^n$, then $$\frac{dy}{dx} = n[u(x)]^{n-1} \cdot \frac{du}{dx}.$$ 3. **Apply the chain rule:** Here, $u(x) = 3x+1$ and $n=5$. 4. Compute $\frac{du}{dx}$: $$\frac{d}{dx}(3x+1) = 3.$$ 5. Substitute into the formula: $$\frac{dy}{dx} = 5(3x+1)^{4} \cdot 3 = 15(3x+1)^4.$$ 6. **Check options:** - A) $5(3x+1)^4$ (missing factor 3) - B) $53(3x+1)^4 x^{-2/3}$ (incorrect and unrelated) - C) $5(3x+1)^4 3x$ (incorrect multiplication by $x$) - D) $13(3x+1)^4 x^{-2/3}$ (incorrect) - E) none of the above 7. The correct derivative is $15(3x+1)^4$, which is not listed exactly. Therefore, the answer is E) none of the above. --- **Final answer:** $$\frac{dy}{dx} = 15(3x+1)^4.$$