Home›Calculus 1›Chain rule›Problem 2.115

Derivative of \( \displaystyle 3 \ln{\left(x + 1 \right)} \)

Problem 2.115 · medium Mental math Video

Differentiate \( \displaystyle f(x) = 3 \ln{\left(x + 1 \right)} \).

\[ \frac{d}{d x} 3 \ln{\left(x + 1 \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 3 \frac{d}{d x} \ln{\left(x + 1 \right)} \]
constant-multiplePull the constant out of the derivative.✓ Proved
\[ = \frac{3}{x + 1} \]
derivative simplifyApply the derivative rule for the natural logarithm. Simplify the expression.✓ Proved

Answer \( \frac{3}{x + 1} \)

Top: the graph of f = 3log(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3/(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.

✓ 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 + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: fail (error) — Step 3 applies the chain rule to differentiate log(x+1) but is labeled 'derivative'. The label 'derivative' is reserved for basic forms (like d/dx log(x)), whereas this step requires the 'chain' rule to handle the inner function (x+1).

Senior review claude-sonnet-5-5, 2026-10-04: pass — The solution is correct: each line is equivalent to the previous one, each step changes one thing, and the labels are legitimate. SymPy proved every line, and the stated answer matches.

qwen3.6:27b-mlx: dismiss — The step is correct and its label is acceptable: the derivative of log(x+1) is 1/(x+1) because the inner function x+1 has derivative 1, which counts as part of the step. The contract does not reserve 'derivative' for basic forms, and when two labels fit, either is acceptable.

Every verdict on record (16)

claude-sonnet-5-5: pass 2026-10-04 — The solution is correct: each line is equivalent to the previous one, each step changes one thing, and the labels are legitimate. SymPy proved every line, and the stated answer matches.
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies the chain rule to differentiate log(x+1) but is labeled 'derivative'. The label 'derivative' is reserved for basic forms (like d/dx log(x)), whereas this step requires the 'chain' rule to handle the inner function (x+1).
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'derivative' but applies the chain rule to differentiate log(x+1); the 'derivative' label is reserved for basic forms like log(x), not composite functions. This is a mislabeling of the chain rule.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 claims to apply the 'derivative' rule to obtain 1/(x+1), but it fails to apply the chain rule for the inner function (x+1). The derivative of log(u) is u'/u; omitting the derivative of the inner term is a mathematical error.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 is labeled 'derivative' but fails to apply the chain rule for the inner function (x+1). The derivative of log(x+1) is 1/(x+1) * d/dx(x+1), so the step skips the chain rule application entirely.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 3 applies the chain rule to differentiate log(x+1) but labels the step as 'derivative'. The rule 'derivative' is reserved for basic forms (like d/dx log(x)), whereas this step requires the 'chain' rule to handle the inner function (x+1). Applying two rules (derivative of log and chain rule) in one step labeled only 'derivative' violates the single-rule constraint and mislabels the operation.
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.