Derivative of \( \displaystyle \frac{2 \sqrt{\left(3 x - 1\right)^{2} + 1}}{3} \)
Problem 2.86 · medium
Differentiate \( \displaystyle f(x) = \frac{2 \sqrt{\left(3 x - 1\right)^{2} + 1}}{3} \).
- \[ \frac{d}{d x} \frac{2 \sqrt{\left(3 x - 1\right)^{2} + 1}}{3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \sqrt{\left(3 x - 1\right)^{2} + 1}}{3} \]constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\left(3 x - 1\right)^{2} + 1\right)}{3 \sqrt{\left(3 x - 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(3 x - 1\right)^{2}}{3 \sqrt{\left(3 x - 1\right)^{2} + 1}} \]algebraSimplify the constant coefficients.✓ Proved
- \[ = \frac{\left(6 x - 2\right) \frac{d}{d x} \left(3 x - 1\right)}{3 \sqrt{\left(3 x - 1\right)^{2} + 1}} \]powerApply the power rule to the inner term.✓ Proved
- \[ = \frac{18 x - 6}{3 \sqrt{\left(3 x - 1\right)^{2} + 1}} \]chain algebraApply the chain rule to the innermost term. Multiply the constants.✓ Proved
- \[ = \frac{6 x - 2}{\sqrt{\left(3 x - 1\right)^{2} + 1}} \]algebra simplifySimplify the expression. Rewrite with a positive exponent and radical notation.✓ Proved
Answer \( \frac{2 \left(3 x - 1\right)}{\sqrt{\left(3 x - 1\right)^{2} + 1}} \)
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 - 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 |
| 10 | ✓ 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: fail (error) — Step 5 applies two rules at once: it simplifies the constant factor (constant‑multiple) and simultaneously drops the derivative of the constant term +1 (derivative of a constant). The label "algebra" is also inappropriate for this step.deepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies two rules at once: it simplifies the constant factor (constant‑multiple) and simultaneously drops the derivative of the constant term +1 (derivative of a constant). The label "algebra" is also inappropriate for this step.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule and constant multiple rule in distinct steps. The labeling is consistent with the provided vocabulary, and the algebraic simplifications are correct.gpt-oss:20b: fail (style) 2026-09-20 — Step 7 incorrectly labels the application of the derivative of the linear term "3*x - 1" as "chain"; it should be labeled "derivative" (or "constant-multiple" if separating the constant factor).qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the chain rule and constant multiple rule in distinct steps. The algebraic simplifications are valid and clearly labeled.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the chain rule and algebraic simplifications in distinct steps. The labels used are appropriate for the operations performed, and the final result matches the stated answer.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 5 incorrectly drops the derivative of the constant term; it applies both the derivative of a sum and the derivative of a constant in one step. Step 7 also mislabels the derivative of a linear function as a chain rule, when it is simply a constant derivative.gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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.