Derivative of \( \displaystyle x - \left(x + 2\right) \ln{\left(x + 2 \right)} \)
Problem 2.1785 · hard
Differentiate \( \displaystyle f(x) = x - \left(x + 2\right) \ln{\left(x + 2 \right)} \).
- \[ \frac{d}{d x} \left(x - \left(x + 2\right) \ln{\left(x + 2 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x - \frac{d}{d x} \left(x + 2\right) \ln{\left(x + 2 \right)} \]sumApply the difference rule.✓ Proved
- \[ = - \left(x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} - \ln{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right) + \frac{d}{d x} x \]productApply the product rule to the second term.✓ Proved
- \[ = - \left(x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} - \ln{\left(x + 2 \right)} + \frac{d}{d x} x \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = - \ln{\left(x + 2 \right)} + \frac{d}{d x} x - 1 \]derivative algebraDifferentiate the logarithmic term. Simplify the fraction.✓ Proved
- \[ = - \ln{\left(x + 2 \right)} \]derivative algebra simplifyDifferentiate the remaining terms. Distribute the negative sign. Combine like terms.✓ Proved
Answer \( - \ln{\left(x + 2 \right)} \)
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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 9 | ✓ 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: fail (error) — Step 7 incorrectly applies the "derivative" rule to an already simplified expression. The expression 1-(log(x+2)+1) is not differentiated; it should simply be simplified to 1-log(x+2)-1. This mislabeling introduces a logical error in the step sequence.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint. All labels are appropriate for the operations performed.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 7 incorrectly applies the "derivative" rule to an already simplified expression. The expression 1-(log(x+2)+1) is not differentiated; it should simply be simplified to 1-log(x+2)-1. This mislabeling introduces a logical error in the step sequence.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint. All labels are appropriate for the operations performed.gpt-oss:20b: fail (style) 2026-10-07 — Step 7 repeats the expression without applying any rule; it should be an algebraic simplification, not a derivative, and the label is incorrect.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
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 2026-10-07 with SymPy 1.14.0.