Derivative of \( \displaystyle - \frac{\sqrt{\left(3 x + 1\right)^{2} + 1}}{3} \)
Problem 2.629 · medium Video
- \[ \frac{d}{d x} \left(- \frac{\sqrt{\left(3 x + 1\right)^{2} + 1}}{3}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \sqrt{\left(3 x + 1\right)^{2} + 1}}{3} \]constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\left(3 x + 1\right)^{2} + 1\right)}{6 \sqrt{\left(3 x + 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(3 x + 1\right)^{2}}{6 \sqrt{\left(3 x + 1\right)^{2} + 1}} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = - \frac{\left(6 x + 2\right) \frac{d}{d x} \left(3 x + 1\right)}{6 \sqrt{\left(3 x + 1\right)^{2} + 1}} \]powerApply the power rule to the squared term.✓ Proved
- \[ = - \frac{18 x + 6}{6 \sqrt{\left(3 x + 1\right)^{2} + 1}} \]derivative constant-multipleDifferentiate the linear term. Multiply the constants.✓ Proved
- \[ = \frac{- 3 x - 1}{\sqrt{\left(3 x + 1\right)^{2} + 1}} \]simplifySimplify the expression.✓ Proved
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify
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 + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**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 + 1)**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: passqwen3.6:27b-mlx: pass
Senior review claude-sonnet-5-5, 2026-10-11: pass — The derivation is correct and each line is proved equal to the previous one. Labels are acceptable; folding -1/3*1/2 into -1/6 during the sum step is trivial arithmetic, not an independent change.
unnamed panel model: dismiss — The only panel entry is a note that one model both passed and objected, with no specific defect stated, and SymPy proved every line and the final answer.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-11 — The derivation is correct and each line is proved equal to the previous one. Labels are acceptable; folding -1/3*1/2 into -1/6 during the sum step is trivial arithmetic, not an independent change.qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies the sum rule but omits the derivative of the constant term +1. The correct derivative of ((3*x+1)**2 + 1) is 2*(3*x+1)*3 + 0, not just 2*(3*x+1)*3.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.