Derivative of \( \displaystyle - 5 x^{2} + \frac{5 x}{2} + \left(10 x^{2} - 5 x + \frac{5}{8}\right) \ln{\left(4 x - 1 \right)} \)
Problem 2.1985 · hard
Differentiate \( \displaystyle f(x) = - 5 x^{2} + \frac{5 x}{2} + \left(10 x^{2} - 5 x + \frac{5}{8}\right) \ln{\left(4 x - 1 \right)} \).
- \[ \frac{d}{d x} \left(- 5 x^{2} + \frac{5 x}{2} + \left(10 x^{2} - 5 x + \frac{5}{8}\right) \ln{\left(4 x - 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} + \frac{d}{d x} \left(- 5 x^{2}\right) + \frac{d}{d x} \left(10 x^{2} - 5 x + \frac{5}{8}\right) \ln{\left(4 x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(10 x^{2} - 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} \left(10 x^{2} - 5 x + \frac{5}{8}\right) + \frac{d}{d x} \frac{5 x}{2} + \frac{d}{d x} \left(- 5 x^{2}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(10 x^{2} - 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \left(\frac{d}{d x} \frac{5}{8} - \frac{d}{d x} 5 x + \frac{d}{d x} 10 x^{2}\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{5 x}{2} + \frac{d}{d x} \left(- 5 x^{2}\right) \]sumApply the sum rule to the derivative of the polynomial.✓ Proved
- \[ = \left(- \frac{d}{d x} 5 x + \frac{d}{d x} 10 x^{2}\right) \ln{\left(4 x - 1 \right)} + \left(10 x^{2} - 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{5 x}{2} + \frac{d}{d x} \left(- 5 x^{2}\right) \]constantThe derivative of the constant 5/8 is 0.✓ Proved
- \[ = \left(\frac{d}{d x} 10 x^{2} - 5\right) \ln{\left(4 x - 1 \right)} + \left(10 x^{2} - 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{5 x}{2} + \frac{d}{d x} \left(- 5 x^{2}\right) \]powerDifferentiate 5*x.✓ Proved
- \[ = \left(20 x - 5\right) \ln{\left(4 x - 1 \right)} + \left(10 x^{2} - 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{5 x}{2} + \frac{d}{d x} \left(- 5 x^{2}\right) \]powerDifferentiate 10*x**2.✓ Proved
- \[ = \left(20 x - 5\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{5 x}{2} + \frac{d}{d x} \left(- 5 x^{2}\right) + \frac{\left(10 x^{2} - 5 x + \frac{5}{8}\right) \frac{d}{d x} \left(4 x - 1\right)}{4 x - 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(20 x - 5\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{5 x}{2} + \frac{d}{d x} \left(- 5 x^{2}\right) + \frac{4 \left(10 x^{2} - 5 x + \frac{5}{8}\right)}{4 x - 1} \]powerDifferentiate 4*x - 1.✓ Proved
- \[ = \left(20 x - 5\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{5 x}{2} + \frac{d}{d x} \left(- 5 x^{2}\right) + \frac{40 x^{2} - 20 x + \frac{5}{2}}{4 x - 1} \]algebraMultiply the terms in the last part.✓ Proved
- \[ = - 10 x + \left(20 x - 5\right) \ln{\left(4 x - 1 \right)} + \frac{5}{2} + \frac{40 x^{2} - 20 x + \frac{5}{2}}{4 x - 1} \]derivativeEvaluate the remaining derivatives.✓ Proved
- \[ = - 10 x + \left(20 x - 5\right) \ln{\left(4 x - 1 \right)} + \frac{5}{2} + \frac{80 x^{2} - 40 x + 5}{8 x - 2} \]algebra simplifySimplify the fraction. Final simplification.✓ Proved
Answer \( 5 \left(4 x - 1\right) \ln{\left(4 x - 1 \right)} \)
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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 undefined where 8*x - 2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 8*x - 2 = 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 (style) — Step 9 applies the derivative of 4*x-1 but labels it as "power"; the correct rule would be "constant-multiple" (or simply "derivative").qwen3.6:27b-mlx: fail (error) — The solution fails to simplify the rational term (40*x**2 - 20*x + 5/2)/(4*x - 1) to 10*x - 5/2, which is necessary to cancel the linear terms and arrive at the stated answer. Consequently, the final expression in step 13 is not algebraically equal to the stated answer.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-10 — Step 9 applies the derivative of 4*x-1 but labels it as "power"; the correct rule would be "constant-multiple" (or simply "derivative").qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to simplify the rational term (40*x**2 - 20*x + 5/2)/(4*x - 1) to 10*x - 5/2, which is necessary to cancel the linear terms and arrive at the stated answer. Consequently, the final expression in step 13 is not algebraically equal to the stated answer.gpt-oss:20b: fail (style) 2026-10-10 — Steps 6 and 9 apply the wrong rule labels: differentiating 5*x and 4*x‑1 should use "constant-multiple", not "power". The steps themselves are algebraically correct, but the labeling violates the contract.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The final simplified expression in step 13 is not algebraically equal to the expression in step 12 (the denominator was changed from 2*(4*x-1) to 8*x-2, but the numerator was not scaled by 4 to compensate). Additionally, the stated answer provided in the prompt is incorrect, but the step-by-step derivation itself contains this algebraic error in the final simplification.
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.