Derivative of \( \displaystyle \frac{\sqrt{\left(3 x - 1\right)^{2} + 1}}{3} \)
Problem 2.574 · medium Video
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(3 x - 1\right)^{2} + 1}}{3} \).
- \[ \frac{d}{d x} \frac{\sqrt{\left(3 x - 1\right)^{2} + 1}}{3} \]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 1/3. Rewrite the square root as a power.✓ 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
- \[ = \frac{\frac{d}{d x} \left(3 x - 1\right)^{2}}{6 \sqrt{\left(3 x - 1\right)^{2} + 1}} \]algebraSimplify the constant coefficients.✓ 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 and chain rule to the inner term.✓ Proved
- \[ = \frac{18 x - 6}{6 \sqrt{\left(3 x - 1\right)^{2} + 1}} \]derivative algebraDifferentiate the innermost linear term. Multiply the constants.✓ Proved
- \[ = \frac{3 x - 1}{\sqrt{\left(3 x - 1\right)^{2} + 1}} \]simplifySimplify the final expression.✓ Proved
Answer \( \frac{3 x - 1}{\sqrt{\left(3 x - 1\right)^{2} + 1}} \)
✓ 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: fail (error) — Step 6 is labeled 'power' but applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Additionally, Step 5 is labeled 'algebra' but performs differentiation of the constant term inside the derivative operator (Derivative((3*x - 1)**2 + 1, x) becomes Derivative((3*x - 1)**2, x)), which is a differentiation step, not algebra.
Senior review claude-sonnet-5-5, 2026-10-10: fail — The math is correct and every line is proved. Step 5 combines the coefficient arithmetic with dropping the derivative of the constant 1, and labels both as 'algebra'. These should be separate steps, with the second labelled sum or constant.
qwen3.6:27b-mlx: uphold — The step 6 complaint is not a defect, since the power rule with the inner derivative left as Derivative(3*x-1) is one chain-style step and either label is acceptable. The step 5 complaint is real: it merges 1/3*1/2 into 1/6 and also drops the constant +1 inside the Derivative operator, a sum/constant-rule step, so 'algebra' is the wrong name and two things change.
Every verdict on record (7)
claude-sonnet-5-5: fail (style) 2026-10-10 — The math is correct and every line is proved. Step 5 combines the coefficient arithmetic with dropping the derivative of the constant 1, and labels both as 'algebra'. These should be separate steps, with the second labelled sum or constant.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 6 is labeled 'power' but applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Additionally, Step 5 is labeled 'algebra' but performs differentiation of the constant term inside the derivative operator (Derivative((3*x - 1)**2 + 1, x) becomes Derivative((3*x - 1)**2, x)), which is a differentiation step, not algebra.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 6 is labeled 'power' but applies both the power rule and the chain rule simultaneously, violating the single-rule constraint. Additionally, Step 5 is labeled 'algebra' but performs differentiation (dropping the derivative of the constant 1), which is a differentiation step, not algebra.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 replaces Derivative((3*x-1)**2+1,x) with Derivative((3*x-1)**2,x), dropping the constant 1 and its derivative in a single move, which violates the one‑rule‑per‑step rule. Step 6 also applies both the power rule and an inner chain rule simultaneously. These combined operations make the solution incorrect under the contract.qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 5 is labeled 'algebra' but performs differentiation (dropping the derivative of the constant 1), which should be labeled 'derivative' or 'sum'.gpt-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.