Derivative of \( \displaystyle - x^{2} + 3 x + \left(2 x^{2} - 6 x + \frac{9}{2}\right) \ln{\left(2 x - 3 \right)} \)
Problem 2.1575 · hard
Differentiate \( \displaystyle f(x) = - x^{2} + 3 x + \left(2 x^{2} - 6 x + \frac{9}{2}\right) \ln{\left(2 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(- x^{2} + 3 x + \left(2 x^{2} - 6 x + \frac{9}{2}\right) \ln{\left(2 x - 3 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 3 x + \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} \left(2 x^{2} - 6 x + \frac{9}{2}\right) \ln{\left(2 x - 3 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(2 x^{2} - 6 x + \frac{9}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x^{2} - 6 x + \frac{9}{2}\right) + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- x^{2}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(2 x^{2} - 6 x + \frac{9}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \left(\frac{d}{d x} \frac{9}{2} - \frac{d}{d x} 6 x + \frac{d}{d x} 2 x^{2}\right) \ln{\left(2 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- x^{2}\right) \]sumApply the sum rule to the polynomial term.✓ Proved
- \[ = \left(4 x - 6\right) \ln{\left(2 x - 3 \right)} + \left(2 x^{2} - 6 x + \frac{9}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- x^{2}\right) \]derivativeDifferentiate the polynomial components.✓ Proved
- \[ = \left(4 x - 6\right) \ln{\left(2 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- x^{2}\right) + \frac{\left(2 x^{2} - 6 x + \frac{9}{2}\right) \frac{d}{d x} \left(2 x - 3\right)}{2 x - 3} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(4 x - 6\right) \ln{\left(2 x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- x^{2}\right) + \frac{2 \left(2 x^{2} - 6 x + \frac{9}{2}\right)}{2 x - 3} \]derivativeDifferentiate the inner function 2*x - 3.✓ Proved
- \[ = - 2 x + \left(4 x - 6\right) \ln{\left(2 x - 3 \right)} + 3 + \frac{2 \left(2 x^{2} - 6 x + \frac{9}{2}\right)}{2 x - 3} \]derivativeEvaluate the remaining derivatives.✓ Proved
- \[ = - 2 x + \left(4 x - 6\right) \ln{\left(2 x - 3 \right)} + 3 + \frac{4 x^{2} - 12 x + 9}{2 x - 3} \]algebraDistribute the 2 in the last term.✓ Proved
- \[ = \left(4 x - 6\right) \ln{\left(2 x - 3 \right)} \]algebra simplify simplifyRecognize the perfect square trinomial. Simplify the fraction. Combine all terms.✓ Proved
Answer \( 2 \left(2 x - 3\right) \ln{\left(2 x - 3 \right)} \)
✓ 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 |
| 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 |
| 12 | ✓ 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. The algebraic simplifications in the final steps are valid and correctly lead to the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications in the final steps are valid and correctly lead to the stated answer.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies differentiation rules and algebraic simplifications in single steps. The final result matches the stated answer.gpt-oss:20b: pass 2026-10-05
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.