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

Problem 2.17 · medium

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

\[ \frac{d}{d x} \frac{\ln{\left(3 x - 1 \right)}}{3} \]
constant-multipleStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(3 x - 1 \right)}}{3} \]
Pull out the constant factor.✓ Proved
\[ = \frac{\frac{d}{d x} \left(3 x - 1\right)}{3 \left(3 x - 1\right)} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \frac{1}{3 x - 1} \]
derivative simplifyDifferentiate the inner function 3*x - 1. Simplify the expression by canceling the 3.✓ Proved

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

Mind the domain. The answer is also defined on (-oo, 1/3), 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.

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 3*x - 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 3*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: fail (style) — Step 2 is missing a rule label; it should be labeled 'constant-multiple'.
qwen3.6:27b-mlx: fail (style) — Step 2 is marked as 'unlabelled', but the operation performed (pulling out a constant factor) is explicitly covered by the 'constant-multiple' rule in the allowed vocabulary. The label should have been 'constant-multiple'.

Every verdict on record (15)

qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 is marked as 'unlabelled', but the operation performed (pulling out a constant factor) is explicitly covered by the 'constant-multiple' rule in the allowed vocabulary. The label should have been 'constant-multiple'.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 is marked as 'unlabelled', but the contract requires every step to name its rule from the fixed vocabulary; it should be labeled 'constant-multiple'.
gpt-oss:20b: fail (style) 2026-09-20 — Step 2 is labeled "unlabelled" which is not an allowed rule name. It should be labeled "constant-multiple" (or "product" if viewed as a constant times a function).
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: fail (style) 2026-09-19 — Step 2 is missing a rule label; it should be labeled 'constant-multiple'.
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 2 is marked as 'unlabelled', but the contract requires every step to name its rule from the fixed vocabulary (it should be 'constant-multiple'). Additionally, Step 1 is labeled 'constant-multiple' but performs no transformation, which violates the 'one change per step' principle; it should likely be labeled 'derivative' or removed.
deepseek-r1:70b: fail (style) 2026-09-19 — Step 2 is missing a rule label; should be 'constant-multiple'.
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (style) 2026-09-18 — Step 2 is marked as 'unlabelled', but the contract requires every step to name its rule from the fixed vocabulary (it should be 'constant-multiple'). Step 1 is also defective because it merely states the problem rather than applying a rule, and it mislabels the starting state as an application of 'constant-multiple'.
deepseek-r1:70b: fail (style) 2026-09-18 — Step 2 is missing a rule label; it should be labelled 'constant-multiple'.
gpt-oss:20b: fail (style) 2026-09-18 — Step 2 uses an unlabelled rule; it should be labeled "constant-multiple" to match the vocabulary and maintain granularity.
gpt-oss:20b: pass 2026-09-16
deepseek-r1:70b: fail 2026-09-16 — Step 2 lacks a rule label, which might confuse students about justifying pulling out the constant.

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.