Derivative of \( \displaystyle 3 x \ln{\left(5 x - 1 \right)} - 3 x - \frac{3 \ln{\left(5 x - 1 \right)}}{5} \)
Problem 2.1102 · hard
Differentiate \( \displaystyle f(x) = 3 x \ln{\left(5 x - 1 \right)} - 3 x - \frac{3 \ln{\left(5 x - 1 \right)}}{5} \).
- \[ \frac{d}{d x} \left(3 x \ln{\left(5 x - 1 \right)} - 3 x - \frac{3 \ln{\left(5 x - 1 \right)}}{5}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(5 x - 1 \right)} - \frac{d}{d x} \frac{3 \ln{\left(5 x - 1 \right)}}{5} \]sumApply the sum rule to separate the terms.✓ Proved
- \[ = \frac{d}{d x} 3 x \ln{\left(5 x - 1 \right)} - \frac{d}{d x} \frac{3 \ln{\left(5 x - 1 \right)}}{5} - 3 \]constantDifferentiate the linear term 3*x.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(5 x - 1 \right)} + 3 \ln{\left(5 x - 1 \right)} - \frac{d}{d x} \frac{3 \ln{\left(5 x - 1 \right)}}{5} - 3 \]productApply the product rule to the first term.✓ Proved
- \[ = \frac{3 x \frac{d}{d x} \left(5 x - 1\right)}{5 x - 1} + 3 \ln{\left(5 x - 1 \right)} - \frac{d}{d x} \frac{3 \ln{\left(5 x - 1 \right)}}{5} - 3 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{15 x}{5 x - 1} + 3 \ln{\left(5 x - 1 \right)} - \frac{d}{d x} \frac{3 \ln{\left(5 x - 1 \right)}}{5} - 3 \]derivative algebraDifferentiate the inner function 5*x - 1. Simplify the product in the second term.✓ Proved
- \[ = \frac{15 x}{5 x - 1} + 3 \ln{\left(5 x - 1 \right)} - 3 - \frac{3}{5 x - 1} \]constant-multiple algebraPull out the constant 3/5 from the derivative. Simplify the last term.✓ Proved
- \[ = 3 \ln{\left(5 x - 1 \right)} - 3 + \frac{15 x - 3}{5 x - 1} \]algebraCombine the terms with the common denominator 5*x - 1.✓ Proved
- \[ = 3 \ln{\left(5 x - 1 \right)} \]algebra algebra algebra simplifyExpress the constant 3 with the common denominator. Distribute the 3 in the numerator. Combine like terms in the numerator. Simplify the final expression.✓ Proved
Answer \( 3 \ln{\left(5 x - 1 \right)} \)
✓ 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 undefined where 5*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules one at a time and uses valid labels from the fixed vocabulary. The algebraic simplification steps are sound and lead to the correct result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules one at a time and uses valid labels from the fixed vocabulary. The algebraic simplification steps are sound and lead to the correct result.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules one at a time and uses valid labels from the fixed vocabulary. The algebraic simplification steps are sound and lead to the correct final answer.gpt-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.