Derivative of \( \displaystyle \frac{3 \operatorname{atan}{\left(2 x - 1 \right)}}{2} \)
Problem 2.576 · medium
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. 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 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) — 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: fail (error) — 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.
Every verdict on record (4)
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.