Derivative of \( \displaystyle 3 x \ln{\left(3 x + 1 \right)} - 3 x + \ln{\left(3 x + 1 \right)} \)
Problem 2.1440 · hard
Differentiate \( \displaystyle f(x) = 3 x \ln{\left(3 x + 1 \right)} - 3 x + \ln{\left(3 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(3 x \ln{\left(3 x + 1 \right)} - 3 x + \ln{\left(3 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \ln{\left(3 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 3 x \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \ln{\left(3 x + 1 \right)} - 3 \]constantThe derivative of 3*x is 3.✓ Proved
- \[ = \frac{d}{d x} \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} - 3 \]algebraCombine the first and third terms using algebra.✓ Proved
- \[ = \left(3 x + 1\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right) - 3 \]productApply the product rule to the first term.✓ Proved
- \[ = 3 \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \left(3 x + 1\right) - 3 \]derivativeDifferentiate 3*x + 1.✓ Proved
- \[ = 3 \ln{\left(3 x + 1 \right)} \]derivative algebra simplifyDifferentiate 3*x + 1 again. Simplify the expression. Final simplified answer.✓ Proved
Answer \( 3 \ln{\left(3 x + 1 \right)} \)
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 |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in a logical sequence. Each step adheres to the one-change-per-step constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and algebraic simplifications in a logical sequence. Each step adheres to the one-change-per-step constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 4 is mathematically incorrect; it claims that 3*x*log(3*x + 1) + log(3*x + 1) equals (3*x + 1)*log(3*x + 1) inside the derivative operator, but the original expression has a -3*x term which is not included in this combination. The solver incorrectly grouped terms across the subtraction of 3*x, violating the linearity of the derivative and the algebraic structure of the function.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.