Derivative of \( \displaystyle 3 x \ln{\left(3 x - 3 \right)} - 3 x - 3 \ln{\left(x - 1 \right)} \)
Problem 2.1104 · hard
Differentiate \( \displaystyle f(x) = 3 x \ln{\left(3 x - 3 \right)} - 3 x - 3 \ln{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \left(3 x \ln{\left(3 x - 3 \right)} - 3 x - 3 \ln{\left(x - 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 3 x \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 1 \right)} - 3 \]constantDifferentiate the term 3*x.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(3 x - 3 \right)} + \ln{\left(3 x - 3 \right)} \frac{d}{d x} 3 x - \frac{d}{d x} 3 \ln{\left(x - 1 \right)} - 3 \]productApply the product rule to the first term.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(3 x - 3 \right)} + 3 \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 1 \right)} - 3 \]constant-multipleDifferentiate 3*x in the first term.✓ Proved
- \[ = \frac{3 x \frac{d}{d x} \left(3 x - 3\right)}{3 x - 3} + 3 \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 1 \right)} - 3 \]chainApply the chain rule to log(3*x - 3).✓ Proved
- \[ = \frac{9 x}{3 x - 3} + 3 \ln{\left(3 x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 1 \right)} - 3 \]derivativeDifferentiate 3*x - 3.✓ Proved
- \[ = \frac{9 x}{3 x - 3} + 3 \ln{\left(3 x - 3 \right)} - 3 - \frac{3 \frac{d}{d x} \left(x - 1\right)}{x - 1} \]constant-multipleDistribute the constant 3 in the last term.✓ Proved
- \[ = \frac{9 x}{3 x - 3} + 3 \ln{\left(3 x - 3 \right)} - 3 - \frac{3}{x - 1} \]derivative algebra algebraDifferentiate x - 1. Simplify the product term. Factor the denominator 3*x - 3.✓ Proved
- \[ = \frac{3 x}{x - 1} + 3 \ln{\left(3 x - 3 \right)} - 3 - \frac{3}{x - 1} \]simplifyCancel the common factor 3.✓ Proved
- \[ = 3 \ln{\left(3 x - 3 \right)} - 3 + \frac{3 x - 3}{x - 1} \]algebra algebraCombine the fractions with the same denominator. Factor the numerator 3*x - 3.✓ Proved
- \[ = 3 \ln{\left(3 x - 3 \right)} \]simplify simplifySimplify the fraction (x-1)/(x-1). Combine the constant terms.✓ Proved
Answer \( 3 \ln{\left(3 x - 3 \right)} \)
✓ 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 |
| 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 undefined where x - 1 = 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 3*x - 3 = 0 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 3*x - 3 = 0 undefined where x - 1 = 0 |
| 12 | ✓ 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 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 16 | ✓ 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 step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed, and the algebraic simplifications are valid.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed, and the algebraic simplifications are valid.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: pass 2026-09-28
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.