Derivative of \( \displaystyle \left(x + 2\right)^{2} \ln{\left(x + 2 \right)} \)
Problem 2.608 · hard
Differentiate \( \displaystyle f(x) = \left(x + 2\right)^{2} \ln{\left(x + 2 \right)} \).
- \[ \frac{d}{d x} \left(x + 2\right)^{2} \ln{\left(x + 2 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \left(x + 2\right)^{2} \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right)^{2} \]productApply the product rule.✓ Proved
- \[ = x + \left(2 x + 4\right) \ln{\left(x + 2 \right)} + 2 \]chain algebraDifferentiate the individual parts using the chain rule. Simplify the first term.✓ Proved
- \[ = \left(x + 2\right) \left(2 \ln{\left(x + 2 \right)} + 1\right) \]algebraFactor out the common term (x + 2).✓ Proved
Answer \( \left(x + 2\right) \left(2 \log{\left(x + 2 \right)} + 1\right) \)
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 log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies the chain rule twice in one line—once for the derivative of log(x+2) and once for the derivative of (x+2)**2—violating the rule that each step must change only one thing. It should be split into two separate chain rule applications.
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.