Derivative of \( \displaystyle 3 x \ln{\left(4 x - 3 \right)} - 3 x - \frac{9 \ln{\left(4 x - 3 \right)}}{4} \)
Problem 2.1227 · hard
Differentiate \( \displaystyle f(x) = 3 x \ln{\left(4 x - 3 \right)} - 3 x - \frac{9 \ln{\left(4 x - 3 \right)}}{4} \).
- \[ \frac{d}{d x} \left(3 x \ln{\left(4 x - 3 \right)} - 3 x - \frac{9 \ln{\left(4 x - 3 \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{9 \ln{\left(4 x - 3 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 3 x \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{9 \ln{\left(4 x - 3 \right)}}{4} - 3 \]constantThe derivative of 3*x is 3.✓ Proved
- \[ = \frac{d}{d x} 3 x \ln{\left(4 x - 3 \right)} - \frac{9 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} - 3 \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{d}{d x} 3 x \ln{\left(4 x - 3 \right)} - 3 - \frac{9 \frac{d}{d x} \left(4 x - 3\right)}{4 \left(4 x - 3\right)} \]logarithmicApply the derivative rule for the logarithm.✓ Proved
- \[ = \frac{d}{d x} 3 x \ln{\left(4 x - 3 \right)} - 3 - \frac{9}{4 x - 3} \]derivative algebraThe derivative of 4*x - 3 is 4. Simplify the constant multiplication.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} 3 x - 3 - \frac{9}{4 x - 3} \]productApply the product rule to the first term.✓ Proved
- \[ = \frac{3 x \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} + 3 \ln{\left(4 x - 3 \right)} - 3 - \frac{9}{4 x - 3} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \frac{12 x}{4 x - 3} + 3 \ln{\left(4 x - 3 \right)} - 3 - \frac{9}{4 x - 3} \]derivative algebraDifferentiate the inner function of the logarithm. Multiply the terms.✓ Proved
- \[ = 3 \ln{\left(4 x - 3 \right)} - 3 + \frac{12 x - 9}{4 x - 3} \]algebra algebraCombine the fractions. Factor the numerator.✓ Proved
- \[ = 3 \ln{\left(4 x - 3 \right)} \]algebra simplifyCancel the common term in the fraction. Final simplification.✓ Proved
Answer \( 3 \ln{\left(4 x - 3 \right)} \)
✓ 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 undefined where 4*x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 15 | ✓ 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. All labels are appropriate for the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. All labels are appropriate for the operations performed.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules one at a time, with appropriate labels from the fixed vocabulary. Each step is logically sound and algebraically correct.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.