Derivative of \( \displaystyle - \frac{x^{2}}{2} + \frac{3 x}{2} + \left(x^{2} - 3 x + \frac{9}{4}\right) \ln{\left(2 x - 3 \right)} \)
Problem 2.1587 · hard
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + \frac{3 x}{2} + \left(x^{2} - 3 x + \frac{9}{4}\right) \ln{\left(2 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + \frac{3 x}{2} + \left(x^{2} - 3 x + \frac{9}{4}\right) \ln{\left(2 x - 3 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \left(x^{2} - 3 x + \frac{9}{4}\right) \ln{\left(2 x - 3 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(x^{2} - 3 x + \frac{9}{4}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} \frac{d}{d x} \left(x^{2} - 3 x + \frac{9}{4}\right) + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(2 x - 3\right) \ln{\left(2 x - 3 \right)} + \left(x^{2} - 3 x + \frac{9}{4}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]derivativeDifferentiate the polynomial term.✓ Proved
- \[ = \left(2 x - 3\right) \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{\left(x^{2} - 3 x + \frac{9}{4}\right) \frac{d}{d x} \left(2 x - 3\right)}{2 x - 3} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(2 x - 3\right) \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{2 \left(x^{2} - 3 x + \frac{9}{4}\right)}{2 x - 3} \]derivative constant-multipleDifferentiate the inner function 2x - 3. Simplify the derivative of the inner function.✓ Proved
- \[ = - x + \left(2 x - 3\right) \ln{\left(2 x - 3 \right)} + \frac{3}{2} + \frac{2 \left(x^{2} - 3 x + \frac{9}{4}\right)}{2 x - 3} \]derivativeEvaluate the remaining derivatives.✓ Proved
- \[ = - x + \left(2 x - 3\right) \ln{\left(2 x - 3 \right)} + \frac{3}{2} + \frac{2 x^{2} - 6 x + \frac{9}{2}}{2 x - 3} \]algebraDistribute the 2 in the numerator.✓ Proved
- \[ = - x + \left(2 x - 3\right) \ln{\left(2 x - 3 \right)} + \frac{3}{2} + \frac{2 x^{2} - 6 x + 4.5}{2 x - 3} \]simplifyConvert fraction to decimal for consistency.✓ Proved
- \[ = - x + \left(2 x - 3\right) \ln{\left(2 x - 3 \right)} + \frac{3}{2} + \frac{2 x^{2} - 6 x + \frac{9}{2}}{2 x - 3} \]simplifyFinal simplification.✓ Proved
Answer \( \left(2 x - 3\right) \ln{\left(2 x - 3 \right)} \)
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 2*x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0 |
| 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: fail (error) — Step 10 converts 9/2 to 4.5, but Step 11 reverts it back to 9/2 without justification or a corresponding algebraic step, creating an inconsistency in the solution path. Additionally, the final answer in Step 11 is not fully simplified; the rational term (2*x**2 - 6*x + 9/2)/(2*x - 3) simplifies to x - 3/4, which would cancel with the -x + 3/2 terms to yield the stated answer (2*x - 3)*log(2*x - 3), but the solution stops short of this cancellation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 10 converts 9/2 to 4.5, but Step 11 reverts it back to 9/2 without justification or a corresponding algebraic step, creating an inconsistency in the solution path. Additionally, the final answer in Step 11 is not fully simplified; the rational term (2*x**2 - 6*x + 9/2)/(2*x - 3) simplifies to x - 3/4, which would cancel with the -x + 3/2 terms to yield the stated answer (2*x - 3)*log(2*x - 3), but the solution stops short of this cancellation.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The final answer provided in the task description is incorrect; the derivative of the polynomial part (-x + 3/2) does not cancel with the rational part, so the result should not be just (2*x - 3)*log(2*x - 3). The steps correctly derive the full expression, but the stated answer is wrong.gpt-oss:20b: fail (error) 2026-10-05 — The solution fails to simplify the rational term (2*x**2-6*x+9/2)/(2*x-3) to x-3/2, which cancels with the earlier -x+3/2, leaving the correct derivative (2*x-3)*log(2*x-3).
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 2026-10-05 with SymPy 1.14.0.