Home›Calculus 1›Chain rule›Problem 2.115

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

Problem 2.115 · medium Mental math

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

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
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).

Every verdict on record (15)

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.