Derivative of \( \displaystyle \left(x - 1\right) \operatorname{atan}{\left(x - 1 \right)} \)
Problem 2.489 · medium
Differentiate \( \displaystyle f(x) = \left(x - 1\right) \operatorname{atan}{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \left(x - 1\right) \operatorname{atan}{\left(x - 1 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(x - 1\right) \frac{d}{d x} \operatorname{atan}{\left(x - 1 \right)} + \operatorname{atan}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) \]productApply the product rule.✓ Proved
- \[ = \left(x - 1\right) \frac{d}{d x} \operatorname{atan}{\left(x - 1 \right)} + \operatorname{atan}{\left(x - 1 \right)} \]derivative constantDifferentiate the first term. Simplify the constant multiplier.✓ Proved
- \[ = \frac{x - 1}{\left(x - 1\right)^{2} + 1} + \operatorname{atan}{\left(x - 1 \right)} \]inverse-trigApply the derivative formula for arctan.✓ Proved
- \[ = \frac{x - 1}{x^{2} - 2 x + 2} + \operatorname{atan}{\left(x - 1 \right)} \]algebraExpand the denominator.✓ Proved
- \[ = \frac{x + \left(x^{2} - 2 x + 2\right) \operatorname{atan}{\left(x - 1 \right)} - 1}{x^{2} - 2 x + 2} \]simplifyCombine into a single fraction.✓ Proved
Answer \( \frac{x - 1}{\left(x - 1\right)^{2} + 1} + \operatorname{atan}{\left(x - 1 \right)} \)
✓ 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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 1 = 0 undefined where x**2 - 2*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 - 2*x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (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 4 applies an algebraic simplification (1 × atan(x‑1) → atan(x‑1)), but it is labeled "constant". The correct label from the allowed vocabulary would be "algebra" or "simplify". No mathematical errors otherwise.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 4 applies an algebraic simplification (1 × atan(x‑1) → atan(x‑1)), but it is labeled "constant". The correct label from the allowed vocabulary would be "algebra" or "simplify". No mathematical errors otherwise.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule, chain rule (implicitly within the inverse-trig step), and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-20 — Step 4 incorrectly labels the simplification of 1*atan(x-1) as "constant"; the appropriate label is "simplify". All other steps correctly apply a single rule from the allowed vocabulary.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20
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.