Derivative of \( \displaystyle x \ln{\left(3 x - 3 \right)} - x - \ln{\left(x - 1 \right)} \)
Problem 2.255 · hard Beautiful
Differentiate \( \displaystyle f(x) = x \ln{\left(3 x - 3 \right)} - x - \ln{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \left(x \ln{\left(3 x - 3 \right)} - x - \ln{\left(x - 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(3 x - 3 \right)} - \frac{d}{d x} \ln{\left(x - 1 \right)} \]sumApply the sum rule for differentiation.✓ Proved
- \[ = \frac{d}{d x} x \ln{\left(3 x - 3 \right)} - \frac{d}{d x} \ln{\left(x - 1 \right)} - 1 \]constantThe derivative of x is 1.✓ Proved
- \[ = x \frac{d}{d x} \ln{\left(3 x - 3 \right)} + \ln{\left(3 x - 3 \right)} \frac{d}{d x} x - \frac{d}{d x} \ln{\left(x - 1 \right)} - 1 \]productApply the product rule to the first term.✓ Proved
- \[ = x \frac{d}{d x} \ln{\left(3 x - 3 \right)} + \ln{\left(3 x - 3 \right)} - \frac{d}{d x} \ln{\left(x - 1 \right)} - 1 \]constantThe derivative of x is 1.✓ Proved
- \[ = \frac{x \frac{d}{d x} \left(3 x - 3\right)}{3 x - 3} + \ln{\left(3 x - 3 \right)} - \frac{d}{d x} \ln{\left(x - 1 \right)} - 1 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{3 x}{3 x - 3} + \ln{\left(3 x - 3 \right)} - \frac{d}{d x} \ln{\left(x - 1 \right)} - 1 \]derivative algebraThe derivative of 3x - 3 is 3. Factor out 3 from the denominator.✓ Proved
- \[ = \frac{x}{x - 1} + \ln{\left(3 x - 3 \right)} - \frac{d}{d x} \ln{\left(x - 1 \right)} - 1 \]simplifySimplify the fraction.✓ Proved
- \[ = \frac{x}{x - 1} + \ln{\left(3 x - 3 \right)} - 1 - \frac{1}{x - 1} \]chainApply the chain rule to the last term.✓ Proved
- \[ = \ln{\left(3 x - 3 \right)} \]algebra simplify simplify algebraCombine the fractions with the constant -1. Simplify the fraction (x-1)/(x-1). The 1 and -1 cancel out. Distribute the 3 inside the logarithm.✓ Proved
Answer \( \log{\left(3 x - 3 \right)} \)
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 undefined where 3*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 3 = 0 undefined where x - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 14 | ✓ 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate for the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 is labeled 'constant' but computes the derivative of x (which is a power rule application), and Step 5 is labeled 'constant' but computes the derivative of x again. The label 'constant' is reserved for the derivative of a constant value, not for differentiating the variable x.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Steps 3 and 5 incorrectly label the derivative of x as a constant rule; the correct label is "derivative". No other steps violate the contract.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — The solution mislabels the rule in steps 3 and 5 as 'constant' when it is actually the derivative of x (an identity rule).deepseek-r1:70b: fail 2026-09-17 — Step 14 incorrectly factors out 3 inside the logarithm, altering the answer unnecessarily.
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.