Derivative of \( \displaystyle - \ln{\left(x + 2 \right)} \)
Problem 2.719 · easy Mental math
Differentiate \( \displaystyle f(x) = - \ln{\left(x + 2 \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(x + 2 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(x + 2 \right)} \]constantPull the constant factor out of the derivative.✓ Proved
- \[ = - \frac{1}{x + 2} \]derivativeApply the derivative rule for the natural logarithm.✓ Proved
Answer \( - \frac{1}{x + 2} \)
Mind the domain. The answer is also defined on (-oo, -2), 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x + 2 = 0 |
| 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: fail (error) — Step 3 is labeled 'derivative', which implies unfolding d/dx on a known form (like log(x)). However, the argument is (x + 2), so the chain rule is required to obtain 1/(x+2). Applying only the basic derivative rule to a composite function without invoking the chain rule is a mathematical error in the step's logic.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is labeled 'derivative', which implies unfolding d/dx on a known form (like log(x)). However, the argument is (x + 2), so the chain rule is required to obtain 1/(x+2). Applying only the basic derivative rule to a composite function without invoking the chain rule is a mathematical error in the step's logic.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 claims to apply the 'derivative' rule to log(x+2), but it fails to apply the chain rule to the inner function (x+2). While the final answer is correct because the derivative of (x+2) is 1, the step skips a necessary rule application, violating the 'one rule per step' constraint.gpt-oss:20b: pass 2026-09-21
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.