Derivative of \( \displaystyle \frac{2 \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \)
Problem 2.199 · medium
Differentiate \( \displaystyle f(x) = \frac{2 \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \).
- \[ \frac{d}{d x} \frac{2 \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \]constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\left(3 x + 2\right)^{2} + 1\right)}{3 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]chain algebraApply the chain rule. Simplify the constant coefficients.✓ Proved
- \[ = \frac{\left(6 x + 4\right) \frac{d}{d x} \left(3 x + 2\right)}{3 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]powerApply the power rule to the inner term.✓ Proved
- \[ = \frac{18 x + 12}{3 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]chain algebraApply the chain rule to the innermost term. Multiply the constants.✓ Proved
- \[ = \frac{6 x + 4}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \]algebra rewriteSimplify the expression. Rewrite the negative exponent as a denominator.✓ Proved
Answer \( \frac{2 \left(3 x + 2\right)}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \)
✓ 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.
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: fail (style) — Step 7 labels the application of the derivative of 3*x+2 as "chain"; the correct label is "derivative" (or "constant-multiple" if one prefers).deepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and algebraic simplifications in distinct steps. All labels accurately reflect the operations performed.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule and algebraic simplifications in distinct steps. All labels accurately reflect the operations performed.gpt-oss:20b: fail (style) 2026-09-20 — Step 7 labels the application of the derivative of 3*x+2 as "chain"; the correct label is "derivative" (or "constant-multiple" if one prefers).qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule and constant multiple rule in distinct steps. The algebraic simplifications and rewrites are valid and clearly labeled.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the chain rule and constant multiple rule in distinct steps. All labels accurately describe the transformations performed, and the algebraic simplifications are correct.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (style) 2026-09-18 — Step 7 is labeled "chain" but it is simply the derivative of the linear function 3*x+2, which should be labeled "derivative". The step itself is correct, but the rule name is inaccurate.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.