Derivative of \( \displaystyle \ln{\left(x + 1 \right)} \)
Problem 2.1021 · easy Mental math
Differentiate \( \displaystyle f(x) = \ln{\left(x + 1 \right)} \).
- \[ \frac{d}{d x} \ln{\left(x + 1 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{1}{x + 1} \]derivativeApply the derivative rule for the natural logarithm.✓ Proved
Answer \( \frac{1}{x + 1} \)
Mind the domain. The answer is also defined on (-oo, -1), where f(x) is not. Substituting there gives a number that is not a slope of f.
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 undefined where x + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x + 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 1 uses an unlabelled rule; it should be a "rewrite" step that simply restates the function as the object of differentiation. The subsequent derivative step is correct, but the first step violates the rule‑labeling requirement.qwen3.6:27b-mlx: fail (error) — The step from log(x + 1) to 1/(x + 1) applies the chain rule implicitly but is labeled only as 'derivative'. The vocabulary requires 'chain' for composite functions, and applying two rules (derivative of log and chain rule) in one step violates the single-rule constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The step from log(x + 1) to 1/(x + 1) applies the chain rule implicitly but is labeled only as 'derivative'. The vocabulary requires 'chain' for composite functions, and applying two rules (derivative of log and chain rule) in one step violates the single-rule constraint.gpt-oss:20b: fail (style) 2026-09-27 — Step 1 uses an unlabelled rule; it should be a "rewrite" step that simply restates the function as the object of differentiation. The subsequent derivative step is correct, but the first step violates the rule‑labeling requirement.qwen3.6:27b-mlx: fail (error) 2026-09-27 — The step from line 1 to line 2 applies the chain rule (differentiating the inner function x+1) but is labeled only as 'derivative'. This violates the one-rule-per-step constraint because the chain rule is not in the allowed vocabulary, and the step performs more than just the basic derivative of log.gpt-oss:20b: fail (style) 2026-09-27 — Step 1 applies the derivative rule but is labeled “unlabelled”; it should be labeled “derivative”. Step 2 then repeats the same result, so the labeling is inconsistent with the required granularity.
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 2026-09-27 with SymPy 1.14.0.