Derivative of \( \displaystyle \frac{5 \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \)
Problem 2.143 · medium
Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \).
- \[ \frac{d}{d x} \frac{5 \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \]constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\left(3 x + 2\right)^{2} + 1\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(3 x + 2\right)^{2}}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]constant-multipleSimplify the constant coefficient.✓ Proved
- \[ = \frac{5 \left(6 x + 4\right) \frac{d}{d x} \left(3 x + 2\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]chainApply the chain rule to the inner squared term.✓ Proved
- \[ = \frac{5 \left(18 x + 12\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]derivative algebraDifferentiate the innermost linear term. Multiply the constants in the derivative term.✓ Proved
- \[ = \frac{15 x + 10}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \]simplify rewriteSimplify the final expression by canceling the 6 and 6/3. Rewrite the negative exponent as a denominator.✓ Proved
Answer \( \frac{5 \left(3 x + 2\right)}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 2)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 2)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 2)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 2)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 2)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 2)**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 2)**2 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (3*x + 2)**2 + 1 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies two changes at once: it pulls out the constant factor (constant‑multiple) and simultaneously drops the derivative of the constant term +1 (an algebraic simplification). Each step must alter only one aspect, so this step violates the contract.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 is missing a rule label, which is required by the contract.gpt-oss:20b: fail (error) 2026-09-19 — Step 5 removes the derivative of the constant term +1 while keeping the same constant factor, effectively applying two rules (constant‑multiple and derivative of a sum) in one step, which violates the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 5 combines two operations—simplifying the derivative of the constant term and applying the constant‑multiple rule—without separating them. Additionally, step 9’s note about “canceling the 6 and 6/3” is misleading; the simplification is simply (5/6)·6 = 5.deepseek-r1:70b: fail 2026-09-17 — Step 9 note incorrectly states the simplification as canceling 6 and 6/3, which is misleading.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-09-26 with SymPy 1.14.0.