Derivative of \( \displaystyle - x \ln{\left(4 x - 3 \right)} + x + \frac{3 \ln{\left(4 x - 3 \right)}}{4} \)
Problem 2.134 · hard
Differentiate \( \displaystyle f(x) = - x \ln{\left(4 x - 3 \right)} + x + \frac{3 \ln{\left(4 x - 3 \right)}}{4} \).
- \[ \frac{d}{d x} \left(- x \ln{\left(4 x - 3 \right)} + x + \frac{3 \ln{\left(4 x - 3 \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- x \ln{\left(4 x - 3 \right)}\right) + \frac{d}{d x} \frac{3 \ln{\left(4 x - 3 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- x \ln{\left(4 x - 3 \right)}\right) + \frac{d}{d x} \frac{3 \ln{\left(4 x - 3 \right)}}{4} + 1 \]derivativeDifferentiate the second term.✓ Proved
- \[ = \frac{d}{d x} \left(- x \ln{\left(4 x - 3 \right)}\right) + \frac{3 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} + 1 \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{d}{d x} \left(- x \ln{\left(4 x - 3 \right)}\right) + 1 + \frac{3 \frac{d}{d x} \left(4 x - 3\right)}{4 \left(4 x - 3\right)} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{d}{d x} \left(- x \ln{\left(4 x - 3 \right)}\right) + 1 + \frac{3}{4 x - 3} \]derivative algebraDifferentiate the inner function 4*x - 3. Simplify the fraction.✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(4 x - 3 \right)} - \ln{\left(4 x - 3 \right)} + 1 + \frac{3}{4 x - 3} \]productApply the product rule to the first term.✓ Proved
- \[ = - \frac{4 x}{4 x - 3} - \ln{\left(4 x - 3 \right)} + 1 + \frac{3}{4 x - 3} \]chain algebraSubstitute the derivative of the log term. Simplify the product term.✓ Proved
- \[ = \frac{3 - 4 x}{4 x - 3} - \ln{\left(4 x - 3 \right)} + 1 \]algebraCombine the fractions with the same denominator.✓ Proved
- \[ = - \ln{\left(4 x - 3 \right)} \]algebra algebra algebra simplifyRewrite the constant 1 with a common denominator. Expand the numerator. Simplify the numerator. Final simplification.✓ Proved
Answer \( - \log{\left(4 x - 3 \right)} \)
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 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 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification in the final steps is sound, leading to the correct result.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification in the final steps is sound, leading to the correct result.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — 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: fail (error) 2026-09-20 — Step 11 incorrectly combines the terms 1 and 3/(4*x-3). The correct combination is 1 + 3/(4*x-3) = 4*x/(4*x-3), which would cancel the -4*x/(4*x-3) term and lead directly to -log(4*x-3). The subsequent steps are based on this incorrect algebra, making the solution mathematically invalid.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules one at a time and uses valid labels from the fixed vocabulary. The algebraic simplifications are sound and lead to the correct result.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification is sound and leads to the correct result.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification in the final steps is valid and leads to the correct result.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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.