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 2.1996 · hard
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{2} - \frac{3 x}{2} + \left(- x^{2} + 3 x - \frac{9}{4}\right) \ln{\left(2 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(\frac{x^{2}}{2} - \frac{3 x}{2} + \left(- x^{2} + 3 x - \frac{9}{4}\right) \ln{\left(2 x - 3 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{d}{d x} \left(- x^{2} + 3 x - \frac{9}{4}\right) \ln{\left(2 x - 3 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(- x^{2} + 3 x - \frac{9}{4}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} \frac{d}{d x} \left(- x^{2} + 3 x - \frac{9}{4}\right) - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{2} \]productApply the product rule to the third term.✓ Proved
- \[ = \left(3 - 2 x\right) \ln{\left(2 x - 3 \right)} + \left(- x^{2} + 3 x - \frac{9}{4}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{2} \]derivativeDifferentiate the polynomial part of the product.✓ Proved
- \[ = \left(3 - 2 x\right) \ln{\left(2 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{\left(- x^{2} + 3 x - \frac{9}{4}\right) \frac{d}{d x} \left(2 x - 3\right)}{2 x - 3} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(3 - 2 x\right) \ln{\left(2 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{2 \left(- x^{2} + 3 x - \frac{9}{4}\right)}{2 x - 3} \]derivativeDifferentiate the inner function 2*x - 3.✓ Proved
- \[ = \left(3 - 2 x\right) \ln{\left(2 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{- 2 x^{2} + 6 x - \frac{9}{2}}{2 x - 3} \]algebraSimplify the expression.✓ Proved
- \[ = x + \left(3 - 2 x\right) \ln{\left(2 x - 3 \right)} - \frac{3}{2} + \frac{- 2 x^{2} + 6 x - \frac{9}{2}}{2 x - 3} \]derivative algebra simplify simplifyEvaluate the derivatives of the first two terms. Distribute the 2 in the last term. Final simplification. Clean up the expression.✓ Proved
Answer \( \left(3 - 2 x\right) \ln{\left(2 x - 3 \right)} \)
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: fail (misleading) — The solution stops short of simplifying the rational term. After dividing (-2*x**2+6*x-9/2)/(2*x-3) one obtains -x+3/2, which cancels with the linear terms, leaving the correct derivative (3-2*x)*log(2*x-3).qwen3.6:27b-mlx: fail (error) — The solution fails to simplify the rational term (-2*x**2 + 6*x - 9/2)/(2*x - 3) to -x + 3/2, which is necessary to cancel the linear terms and arrive at the stated answer. Consequently, the final expression in step 11 is not equal to the stated answer.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-10 — The solution stops short of simplifying the rational term. After dividing (-2*x**2+6*x-9/2)/(2*x-3) one obtains -x+3/2, which cancels with the linear terms, leaving the correct derivative (3-2*x)*log(2*x-3).qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to simplify the rational term (-2*x**2 + 6*x - 9/2)/(2*x - 3) to -x + 3/2, which is necessary to cancel the linear terms and arrive at the stated answer. Consequently, the final expression in step 11 is not equal to the stated answer.gpt-oss:20b: pass 2026-10-10 — The solution stops short of simplifying the rational term, which cancels with the linear terms to give the compact result (3-2*x)*log(2*x-3).qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to simplify the rational term (-2*x**2 + 6*x - 9/2)/(2*x - 3) to -x + 3/2, which is necessary to cancel the polynomial terms and arrive at the stated answer. Consequently, the final expression in step 11 is not equal to the stated answer.
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 2026-10-10 with SymPy 1.14.0.