Derivative of \( \displaystyle x \left(\ln{\left(3 x \right)} - 1\right) \)
Problem 2.265 · hard
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(3 x \right)} - 1\right) \).
- \[ \frac{d}{d x} x \left(\ln{\left(3 x \right)} - 1\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(\ln{\left(3 x \right)} - 1\right) + \left(\ln{\left(3 x \right)} - 1\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = x \frac{d}{d x} \left(\ln{\left(3 x \right)} - 1\right) + \ln{\left(3 x \right)} - 1 \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = - x \frac{d}{d x} 1 + x \frac{d}{d x} \ln{\left(3 x \right)} + \ln{\left(3 x \right)} - 1 \]sumDistribute the derivative across the sum.✓ Proved
- \[ = x \frac{d}{d x} \ln{\left(3 x \right)} + \ln{\left(3 x \right)} - 1 \]derivative algebraDifferentiate the constant term. Simplify the zero term.✓ Proved
- \[ = \ln{\left(3 x \right)} \]chain derivative algebra algebra simplifyApply the chain rule to the logarithm. Differentiate the inner function 3*x. Simplify the fraction. Simplify the product of 1/x and x. Combine the constant terms.✓ Proved
Answer \( \log{\left(3 x \right)} \)
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 |
| 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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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 8 is labeled "derivative" but it only simplifies the factor 3/1 to 3; it should be an "algebra" step. This mislabeling violates the rule that each step must name the exact rule applied.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (style) — Step 4 applies the linearity of the derivative operator (splitting the derivative of a sum) but labels it 'sum', which is not the standard label for differentiation rules in this vocabulary; 'derivative' or 'sum' (if interpreted as linearity) is ambiguous, but more critically, Step 7 and 8 split the chain rule application into two steps with confusing labels. Step 7 introduces the inner derivative factor `3/1` inside the expression but labels it 'chain', while Step 8 simplifies that constant factor but labels it 'derivative'. The chain rule is a single logical step: d/dx log(u) = u'/u. Splitting it into 'chain' (introducing the structure) and 'derivative' (evaluating the inner derivative) violates the 'one rule per step' principle if the rule is 'chain', or mislabels the simplification in step 8. Specifically, Step 7 transforms `Derivative(log(3*x), x)` into `(1/(3*x) * (3/1))*x`? No, Step 7 transforms `Derivative(log(3*x), x)` into `(1/(3*x) * 3)`. The note says 'Apply the chain rule'. The expression in step 7 is `log(3*x) - 1 + (1/(3*x) * (3/1))*x`. This looks like it's trying to show the chain rule structure `f'(g(x)) * g'(x)`. However, Step 8 then simplifies `3/1` to `3` and labels it 'derivative'. Simplifying a constant fraction is 'algebra' or 'simplify', not 'derivative'. This is a labeling defect.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 4 applies the linearity of the derivative operator (splitting the derivative of a sum) but labels it 'sum', which is not the standard label for differentiation rules in this vocabulary; 'derivative' or 'sum' (if interpreted as linearity) is ambiguous, but more critically, Step 7 and 8 split the chain rule application into two steps with confusing labels. Step 7 introduces the inner derivative factor `3/1` inside the expression but labels it 'chain', while Step 8 simplifies that constant factor but labels it 'derivative'. The chain rule is a single logical step: d/dx log(u) = u'/u. Splitting it into 'chain' (introducing the structure) and 'derivative' (evaluating the inner derivative) violates the 'one rule per step' principle if the rule is 'chain', or mislabels the simplification in step 8. Specifically, Step 7 transforms `Derivative(log(3*x), x)` into `(1/(3*x) * (3/1))*x`? No, Step 7 transforms `Derivative(log(3*x), x)` into `(1/(3*x) * 3)`. The note says 'Apply the chain rule'. The expression in step 7 is `log(3*x) - 1 + (1/(3*x) * (3/1))*x`. This looks like it's trying to show the chain rule structure `f'(g(x)) * g'(x)`. However, Step 8 then simplifies `3/1` to `3` and labels it 'derivative'. Simplifying a constant fraction is 'algebra' or 'simplify', not 'derivative'. This is a labeling defect.gpt-oss:20b: fail (style) 2026-09-20 — Step 8 is labeled "derivative" but it only simplifies the factor 3/1 to 3; it should be an "algebra" step. This mislabeling violates the rule that each step must name the exact rule applied.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 4 applies the linearity of the derivative operator (distributing the derivative over a sum), which is best labeled 'sum' or 'algebra', but the note claims to 'distribute the derivative across the sum' while the step also implicitly handles the constant term separation. More critically, Step 7 and 8 split the chain rule application into two steps: Step 7 introduces the derivative of the outer function and the derivative of the inner function simultaneously (or rather, sets up the chain rule structure), but Step 8 is labeled 'derivative' for differentiating the inner function 3x. While splitting chain rule steps is acceptable, Step 7's label 'chain' is appropriate for the whole operation, but Step 8's label 'derivative' is correct for d/dx(3x). However, Step 4 is problematic: it transforms `Derivative(log(3*x) - 1, x)` into `Derivative(log(3*x), x) - Derivative(1, x)`. This is an application of the sum rule (linearity). The label 'sum' is correct. But wait, look at Step 7: `Derivative(log(3*x), x)` becomes `(1/(3*x) * (3/1))`. This step applies the chain rule AND simplifies the derivative of the inner function `3x` to `3` (or `3/1`). The label is 'chain'. Step 8 then takes `(1/(3*x) * (3/1))` and changes it to `(1/(3*x) * 3)`. This is just algebraic simplification of `3/1` to `3`. The label is 'derivative'. This is a defect: Step 8 is an algebraic simplification, not a differentiation step. The differentiation of the inner function was already performed in Step 7 (implicitly or explicitly). Step 8 does not apply a derivative rule; it simplifies a constant fraction. Therefore, the label 'derivative' is incorrect for Step 8.gpt-oss:20b: fail (style) 2026-09-20 — Step 8 is labeled "derivative" but only performs an algebraic simplification of (1/(3*x)*3) to 1/x. The correct label for this step would be "algebra".qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 7 applies two rules at once (chain rule followed by algebraic multiplication), violating the one‑rule‑per‑step rule. Additionally, step 8 is labeled as "derivative" even though it only simplifies the factor 3/1 to 3, an algebraic simplification.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 8 applies a "derivative" label to an algebraic simplification; it should be labeled "algebra" or "simplify". The step also redundantly re‑derives the inner function 3*x, which was already handled in step 7.gpt-oss:20b: fail 2026-09-17 — Step 8 incorrectly labels the simplification of (3/1) as a derivative; the rule should be algebraic simplification. This mislabeling could mislead a student into thinking a derivative is being taken when only a simplification is performed.deepseek-r1:70b: pass 2026-09-17 — The solution is correct with clear steps and appropriate rule applications.
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.