Derivative of \( \displaystyle \operatorname{atan}^{2}{\left(x + 1 \right)} \)
Problem 2.617 · medium
Differentiate \( \displaystyle f(x) = \operatorname{atan}^{2}{\left(x + 1 \right)} \).
- \[ \frac{d}{d x} \operatorname{atan}^{2}{\left(x + 1 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 2 \operatorname{atan}{\left(x + 1 \right)} \frac{d}{d x} \operatorname{atan}{\left(x + 1 \right)} \]powerApply the power rule.✓ Proved
- \[ = \frac{2 \operatorname{atan}{\left(x + 1 \right)}}{\left(x + 1\right)^{2} + 1} \]derivative algebraDifferentiate the inner function using the derivative of arctan. Simplify the expression.✓ Proved
- \[ = \frac{2 \operatorname{atan}{\left(x + 1 \right)}}{x^{2} + 2 x + 2} \]algebra simplifyExpand the denominator. Combine like terms.✓ Proved
Answer \( \frac{2 \operatorname{atan}{\left(x + 1 \right)}}{\left(x + 1\right)^{2} + 1} \)
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 (x + 1)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 1 = 0 |
| 5 | ✓ 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 |
| 6 | ✓ 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: passqwen3.6:27b-mlx: fail (error) — Step 2 applies the power rule but fails to apply the chain rule for the inner function (x+1), incorrectly leaving a Derivative term instead of computing the derivative of the inner function. This violates the 'one rule per step' constraint by conflating the power rule with an incomplete chain rule application.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 applies the power rule but fails to apply the chain rule for the inner function (x+1), incorrectly leaving a Derivative term instead of computing the derivative of the inner function. This violates the 'one rule per step' constraint by conflating the power rule with an incomplete chain rule application.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 2 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a power rule step leaving the derivative operator, followed by a chain rule step.gpt-oss:20b: pass 2026-09-21
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.