Derivative of \( \displaystyle \left(x + 2\right) \operatorname{atan}{\left(x + 2 \right)} \)
Problem 2.559 · medium
Differentiate \( \displaystyle f(x) = \left(x + 2\right) \operatorname{atan}{\left(x + 2 \right)} \).
- \[ \frac{d}{d x} \left(x + 2\right) \operatorname{atan}{\left(x + 2 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(x + 2\right) \frac{d}{d x} \operatorname{atan}{\left(x + 2 \right)} + \operatorname{atan}{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right) \]productApply the product rule.✓ Proved
- \[ = \left(x + 2\right) \frac{d}{d x} \operatorname{atan}{\left(x + 2 \right)} + \operatorname{atan}{\left(x + 2 \right)} \]derivative constantDifferentiate the first part of the product. The derivative of (x + 2) is 1.✓ Proved
- \[ = \frac{x + 2}{\left(x + 2\right)^{2} + 1} + \operatorname{atan}{\left(x + 2 \right)} \]inverse-trig simplifyDifferentiate the arctangent function using the chain rule. Simplify the expression.✓ Proved
Answer \( \frac{x + 2}{\left(x + 2\right)^{2} + 1} + \operatorname{atan}{\left(x + 2 \right)} \)
✓ 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 + 2)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (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 (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 4 incorrectly labels the simplification of 1*atan(x+2) as "constant"; it should be an "algebra" or "simplify" step.qwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'constant' but performs algebraic simplification (1*f -> f); the label 'constant' refers to the derivative of a constant, not multiplication by 1. Step 5 is labeled 'inverse-trig' but applies the chain rule to differentiate atan(x+2); while 'inverse-trig' names the function, the step involves a composition that typically requires 'chain' or a combined label, but given the strict one-rule-per-step constraint, applying the derivative formula for atan(u) where u is not x is a compound step or mislabeled if 'chain' is not used.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 4 is labeled 'constant' but performs algebraic simplification (1*f -> f); the label 'constant' refers to the derivative of a constant, not multiplication by 1. Step 5 is labeled 'inverse-trig' but applies the chain rule to differentiate atan(x+2); while 'inverse-trig' names the function, the step involves a composition that typically requires 'chain' or a combined label, but given the strict one-rule-per-step constraint, applying the derivative formula for atan(u) where u is not x is a compound step or mislabeled if 'chain' is not used.gpt-oss:20b: fail (style) 2026-09-21 — Step 4 incorrectly labels the simplification of 1*atan(x+2) as "constant"; it should be an "algebra" or "simplify" step.qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 applies the derivative rule to the first term but leaves the second term undifferentiated, violating the 'one rule per step' constraint; Step 4 simplifies '1*atan' which is algebra, not a constant rule application.gpt-oss:20b: fail (style) 2026-09-21 — Step 4 incorrectly labels the simplification of 1*atan(x+2) as "constant"; it should be "simplify". Step 5 applies both the inverse‑trig derivative rule and the chain rule in one step, which violates the one‑rule‑per‑step rule and should be split into two separate steps with appropriate labels.
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.