Home›Calculus 1›Chain rule›Problem 2.177

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

Problem 2.177 · medium Mental math

Differentiate \( \displaystyle f(x) = \ln{\left(x + 2 \right)} \).

\[ \frac{d}{d x} \ln{\left(x + 2 \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{1}{x + 2} \]
logarithmicApply the rule for the derivative of 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 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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: fail (error) — Step 2 applies the chain rule implicitly by differentiating the inner function (x+2) to 1, but labels the step 'logarithmic'. The 'logarithmic' rule alone transforms d/dx(log(u)) to u'/u; it does not compute u'. Since the step changes the inner expression's derivative without labeling it 'chain' or 'algebra', it violates the one-rule-per-step constraint.

Every verdict on record (15)

qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 applies the chain rule implicitly by differentiating the inner function (x+2) to 1, but labels the step 'logarithmic'. The 'logarithmic' rule alone transforms d/dx(log(u)) to u'/u; it does not compute u'. Since the step changes the inner expression's derivative without labeling it 'chain' or 'algebra', it violates the one-rule-per-step constraint.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-20 — The step from line 1 to line 2 applies both the logarithmic derivative rule and the chain rule simultaneously. The derivative of log(u) is u'/u; here u = x+2, so u' = 1. A single step cannot combine the outer derivative (logarithmic) and the inner derivative (chain) without violating the 'one change per step' constraint.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 applies the chain rule implicitly but labels it 'logarithmic'. The transition from d/dx(log(x+2)) to 1/(x+2) requires differentiating the inner function (x+2), which is a separate operation from the logarithmic derivative rule. This violates the one-rule-per-step constraint.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 applies the chain rule to differentiate the composite function log(x+2), but is labeled 'logarithmic'. The 'logarithmic' rule alone only handles d/dx(log(x)), so the step violates the single-rule constraint by implicitly applying the chain rule without labeling it.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-18 — The transition from step 1 to step 2 applies the chain rule (differentiating the inner function x+2) but labels it only as 'logarithmic'. This violates the single-rule-per-step constraint because the chain rule is a distinct operation from the logarithmic derivative rule.
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-17
deepseek-r1:70b: pass 2026-09-17

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.