Derivative of \( \displaystyle \left(x + 2\right) \operatorname{atan}{\left(x + 2 \right)} \)
Problem 2.559 · medium Video
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. 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 |
| 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 (error) — Step 5 applies both the derivative of the arctangent (inverse‑trig rule) and the chain rule at once, violating the one‑rule‑per‑step rule. The correct labeling would need two separate steps: one for the derivative of atan and one for the chain rule.qwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to differentiate atan(x+2) but labels it 'inverse-trig'. The 'inverse-trig' rule typically refers to the standard derivative formula for atan(u) where u is the variable, or it is insufficient to describe the composition with (x+2) which requires the chain rule. More critically, Step 3 and 4 are redundant and mislabeled: Step 3 claims to differentiate the first part but leaves the derivative operator on the second part, and Step 4 simplifies 1*atan to atan while claiming the rule is 'constant' (which usually refers to the derivative of a constant being zero, or pulling out constants). The primary defect is Step 5: it performs a chain rule differentiation (derivative of atan(u) * u') but labels it 'inverse-trig', ignoring the chain rule component explicitly mentioned in the note. The vocabulary requires 'chain' for composite functions. Additionally, Step 3 is labeled 'derivative' but only partially evaluates the expression, which is not a standard single-rule application; it should likely be part of the product rule expansion or a separate simplification. However, the most glaring error is Step 5's label mismatch with the operation described in the note (chain rule).
Senior review claude-sonnet-5-5, 2026-10-10: pass — The derivation is correct and every step was checked by SymPy. The 'constant' label on step 4 (dropping the factor 1) is loose, and its note repeats step 3, but this is not a false statement.
gpt-oss:20b: dismiss — Step 5 evaluates d/dx atan(x+2) as 1/(1+(x+2)^2), and the contract counts evaluating a simple inner derivative as part of that one step, so it is a single step.qwen3.6:27b-mlx: dismiss — 'inverse-trig' is a legitimate label for the atan derivative step, and where two labels fit (chain or inverse-trig) either is acceptable. Step 3 changes one term and Step 4 removes the factor 1, both valid, and SymPy checked every line.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — The derivation is correct and every step was checked by SymPy. The 'constant' label on step 4 (dropping the factor 1) is loose, and its note repeats step 3, but this is not a false statement.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 5 applies the chain rule to differentiate atan(x+2) but labels it 'inverse-trig'. The 'inverse-trig' rule typically refers to the standard derivative formula for atan(u) where u is the variable, or it is insufficient to describe the composition with (x+2) which requires the chain rule. More critically, Step 3 and 4 are redundant and mislabeled: Step 3 claims to differentiate the first part but leaves the derivative operator on the second part, and Step 4 simplifies 1*atan to atan while claiming the rule is 'constant' (which usually refers to the derivative of a constant being zero, or pulling out constants). The primary defect is Step 5: it performs a chain rule differentiation (derivative of atan(u) * u') but labels it 'inverse-trig', ignoring the chain rule component explicitly mentioned in the note. The vocabulary requires 'chain' for composite functions. Additionally, Step 3 is labeled 'derivative' but only partially evaluates the expression, which is not a standard single-rule application; it should likely be part of the product rule expansion or a separate simplification. However, the most glaring error is Step 5's label mismatch with the operation described in the note (chain rule).gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies both the derivative of the arctangent (inverse‑trig rule) and the chain rule at once, violating the one‑rule‑per‑step rule. The correct labeling would need two separate steps: one for the derivative of atan and one for the chain rule.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.