Derivative of \( \displaystyle \ln{\left(\frac{e^{3 x}}{3 x} \right)} \)
Problem 2.1138 · hard Beautiful
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{3 x}}{3 x} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\frac{e^{3 x}}{3 x} \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(3 x - \ln{\left(3 x \right)}\right) \]simplify simplifyUse logarithm properties to split the expression. Simplify the log of the exponential term.✓ Proved
- \[ = \frac{d}{d x} 3 x - \frac{d}{d x} \ln{\left(3 x \right)} \]sumApply the difference rule.✓ Proved
- \[ = 3 - \frac{d}{d x} \ln{\left(3 x \right)} \]constant-multipleDifferentiate the first term.✓ Proved
- \[ = 3 - \frac{\frac{d}{d x} 3 x}{3 x} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = 3 - \frac{1}{x} \]derivative algebraDifferentiate the linear term. Simplify the fraction.✓ Proved
- \[ = \frac{3 x - 1}{x} \]simplifyCombine into a single fraction.✓ Proved
Answer \( 3 - \frac{1}{x} \)
Mind the domain. The answer is also defined on (-oo, 0), 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 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 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 a rule from the allowed vocabulary.qwen3.6:27b-mlx: fail (error) — Step 5 is labeled 'constant-multiple' but performs the differentiation of the first term (derivative of 3x is 3). The rule 'constant-multiple' factors out constants, it does not compute derivatives. This is a mislabeling of the operation performed.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 5 is labeled 'constant-multiple' but performs the differentiation of the first term (derivative of 3x is 3). The rule 'constant-multiple' factors out constants, it does not compute derivatives. This is a mislabeling of the operation performed.gpt-oss:20b: fail (style) 2026-09-28 — Step 1 lacks a rule label; every step must name a rule from the allowed vocabulary.qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies logarithmic properties, the sum rule, and the chain rule. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (style) 2026-09-28 — Step 5 incorrectly labels the differentiation of 3*x as "constant-multiple"; it should be "derivative". The rest of the steps are fine.
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 2026-09-28 with SymPy 1.14.0.