Derivative of \( \displaystyle x^{2} - \frac{x}{2} + \left(- 2 x^{2} + x - \frac{1}{8}\right) \ln{\left(4 x - 1 \right)} \)
Problem 2.2124 · hard
Differentiate \( \displaystyle f(x) = x^{2} - \frac{x}{2} + \left(- 2 x^{2} + x - \frac{1}{8}\right) \ln{\left(4 x - 1 \right)} \).
- \[ \frac{d}{d x} \left(x^{2} - \frac{x}{2} + \left(- 2 x^{2} + x - \frac{1}{8}\right) \ln{\left(4 x - 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} x^{2} + \frac{d}{d x} \left(- 2 x^{2} + x - \frac{1}{8}\right) \ln{\left(4 x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(- 2 x^{2} + x - \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} \left(- 2 x^{2} + x - \frac{1}{8}\right) - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} x^{2} \]productApply the product rule to the third term.✓ Proved
- \[ = \left(1 - 4 x\right) \ln{\left(4 x - 1 \right)} + \left(- 2 x^{2} + x - \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} x^{2} \]derivativeDifferentiate the polynomial part.✓ Proved
- \[ = \left(1 - 4 x\right) \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} x^{2} + \frac{\left(- 2 x^{2} + x - \frac{1}{8}\right) \frac{d}{d x} \left(4 x - 1\right)}{4 x - 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(1 - 4 x\right) \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} x^{2} + \frac{4 \left(- 2 x^{2} + x - \frac{1}{8}\right)}{4 x - 1} \]derivativeDifferentiate the inner function 4x - 1.✓ Proved
- \[ = \left(1 - 4 x\right) \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} x^{2} + \frac{- 8 x^{2} + 4 x - \frac{1}{2}}{4 x - 1} \]algebraSimplify the expression.✓ Proved
- \[ = 2 x + \left(1 - 4 x\right) \ln{\left(4 x - 1 \right)} - \frac{1}{2} + \frac{- 8 x^{2} + 4 x - \frac{1}{2}}{4 x - 1} \]derivative algebraEvaluate the derivatives of the first two terms. Distribute the 4 in the last term.✓ Proved
- \[ = 2 x + \left(1 - 4 x\right) \ln{\left(4 x - 1 \right)} - \frac{1}{2} + \frac{- 16 x^{2} + 8 x - 1}{8 x - 2} \]algebra algebraSimplify the fraction. Expand the denominator.✓ Proved
- \[ = 2 x + \left(1 - 4 x\right) \ln{\left(4 x - 1 \right)} - \frac{1}{2} + \frac{- 8 x^{2} + 4 x - \frac{1}{2}}{4 x - 1} \]simplifyFinal simplification of the expression.✓ Proved
Answer \( \left(1 - 4 x\right) \ln{\left(4 x - 1 \right)} \)
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 undefined where 4*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 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 undefined where 8*x - 2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 8*x - 2 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 8*x - 2 = 0 undefined where 4*x - 1 = 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: pass — The solution correctly applies differentiation rules step-by-step. The final expression is algebraically equivalent to the stated answer, as the rational term simplifies to 1/2, which cancels the -1/2 constant.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies differentiation rules step-by-step. The final expression is algebraically equivalent to the stated answer, as the rational term simplifies to 1/2, which cancels the -1/2 constant.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to simplify the rational term (-8*x**2 + 4*x - 1/2)/(4*x - 1) to 1/2, which is necessary to cancel the -1/2 from the polynomial part and arrive at the stated answer (1 - 4*x)*log(4*x - 1). The final expression in step 12 is algebraically unequal 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 2026-10-12 with SymPy 1.14.0.