Home›Calculus 1›Derivatives of logarithms›Problem 2.719

Derivative of \( \displaystyle - \ln{\left(x + 2 \right)} \)

Problem 2.719 · easy Mental math Video

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 \( - \frac{1}{x + 2} \)

Top: the graph of f = -log(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -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.

✓ 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 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: fail (style) — Step 1 lacks a rule label; every step must name the rule applied, e.g., "derivative".
qwen3.6:27b-mlx: fail (error) — Step 3 applies the derivative of log(x+2) but fails to apply the chain rule for the inner function (x+2). While the derivative of (x+2) is 1, the step skips the chain rule application entirely, violating the 'one rule per step' contract by implicitly combining the derivative of the outer function and the inner function.

Senior review claude-sonnet-5-5, 2026-10-12: pass — The solution is correct: -log(x+2) differentiates to -1/(x+2). The constant-multiple step and the log derivative step are each a single change with an acceptable label.

gpt-oss:20b: dismiss — Line 1 only restates the problem as a Derivative expression; it applies no rule, so it needs no label, and SymPy proved it matches the problem.
qwen3.6:27b-mlx: dismiss — The contract says evaluating the derivative of a simple inner function such as x+2 is part of a chain-rule step, so one step giving -1/(x+2) is acceptable; the 'derivative' label is also legitimate, and SymPy proved the line correct.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — The solution is correct: -log(x+2) differentiates to -1/(x+2). The constant-multiple step and the log derivative step are each a single change with an acceptable label.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 applies the derivative of log(x+2) but fails to apply the chain rule for the inner function (x+2). While the derivative of (x+2) is 1, the step skips the chain rule application entirely, violating the 'one rule per step' contract by implicitly combining the derivative of the outer function and the inner function.
gpt-oss:20b: fail (style) 2026-09-26 — Step 1 lacks a rule label; every step must name the rule applied, e.g., "derivative".
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 2026-09-21
qwen3.6:27b-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.