Derivative of \( \displaystyle \frac{3 \operatorname{atan}{\left(2 x - 1 \right)}}{2} \)
Problem 2.576 · medium Video
Differentiate \( \displaystyle f(x) = \frac{3 \operatorname{atan}{\left(2 x - 1 \right)}}{2} \).
- \[ \frac{d}{d x} \frac{3 \operatorname{atan}{\left(2 x - 1 \right)}}{2} \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \operatorname{atan}{\left(2 x - 1 \right)}}{2} \]chainApply the chain rule to the inner function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(2 x - 1\right)}{2 \left(\left(2 x - 1\right)^{2} + 1\right)} \]derivativeDifferentiate the outer function (arctan).✓ Proved
- \[ = \frac{3}{\left(2 x - 1\right)^{2} + 1} \]algebra algebra algebraEvaluate the derivative of the inner linear function. Multiply the constants. Simplify the fraction.✓ Proved
- \[ = \frac{3}{4 x^{2} - 4 x + 2} \]algebra simplify simplifyExpand the squared binomial. Combine like terms. Factor out the common term.✓ Proved
Answer \( \frac{3}{\left(2 x - 1\right)^{2} + 1} \)
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. A reviewer's objection to the explanation stands; see the receipt. 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 undefined where (2*x - 1)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 1)**2 + 1 = 0 undefined where 4*x**2 - 4*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (2*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 (style) — The solution mislabels several steps: step 3 applies the chain rule but is labeled "derivative", and step 4 differentiates the inner linear function but is labeled "algebra". These labeling errors violate the contract’s rule‑granularity requirement.qwen3.6:27b-mlx: fail (style) — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem, violating the rule that each step must change one thing. Step 2 is labeled 'chain' but only factors out the constant 3/2, which is a 'constant-multiple' step, not an application of the chain rule.
Senior review claude-sonnet-5-5, 2026-10-10: fail — The math is correct and every line is verified, but the rule labels are shifted by one step. Line 2 is labeled 'chain' when it is a constant-multiple step, and line 4 evaluates d/dx(2x-1)=2 but is labeled 'algebra'.
gpt-oss:20b: dismiss — Line 3 differentiates arctan and leaves Derivative(2x-1) in place, so 'derivative' is an acceptable label; the claim that it should be 'chain' is not a real defect, and the 'algebra' label on line 4 is only a minor issue.qwen3.6:27b-mlx: uphold — Line 2 only pulls the constant 3/2 out of the derivative, which is constant-multiple, so the 'chain' label names a rule the step did not apply; the objection to line 1 (a restatement) is not a defect.(unnamed panel model that both passed and objected): dismiss — No specific defect is given, and every line is already proven equivalent by SymPy.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-10 — The math is correct and every line is verified, but the rule labels are shifted by one step. Line 2 is labeled 'chain' when it is a constant-multiple step, and line 4 evaluates d/dx(2x-1)=2 but is labeled 'algebra'.qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem, violating the rule that each step must change one thing. Step 2 is labeled 'chain' but only factors out the constant 3/2, which is a 'constant-multiple' step, not an application of the chain rule.gpt-oss:20b: fail (style) 2026-09-26 — The solution mislabels several steps: step 3 applies the chain rule but is labeled "derivative", and step 4 differentiates the inner linear function but is labeled "algebra". These labeling errors violate the contract’s rule‑granularity requirement.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem. Step 2 is labeled 'chain' but only factors out the constant 3/2, which is a 'constant-multiple' rule application, not the chain rule.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels the application of the chain rule as "derivative" and splits the chain rule into two separate sub‑steps. The correct single rule name for that step is "chain" (or simply "derivative" if the outer derivative is treated as a known formula, but it should not be split).qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule, and derivative of arctan in separate steps. The subsequent algebraic simplifications are valid and clearly labeled.gpt-oss:20b: fail (error) 2026-09-21 — Step 2 incorrectly labels the constant‑multiple step as "chain". Step 3 applies both the derivative of atan and the chain rule in one line, violating the rule that each step must change only one thing. These mislabelings and combined operations make the solution incorrect according to the contract.
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.