Derivative of \( \displaystyle x^{2} + \frac{x}{2} - \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)} \)
Problem 2.2427 · hard Video
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
- \[ = - \left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} - \left(\frac{d}{d x} \frac{1}{8} + \frac{d}{d x} x + \frac{d}{d x} 2 x^{2}\right) \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} x^{2} \]sumApply the sum rule to the first part of the product rule.✓ Proved
- \[ = - \left(\frac{d}{d x} x + \frac{d}{d x} 2 x^{2}\right) \ln{\left(4 x + 1 \right)} - \left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} x^{2} \]constantThe derivative of the constant 1/8 is 0.✓ Proved
- \[ = - \left(\frac{d}{d x} x + 2 \frac{d}{d x} x^{2}\right) \ln{\left(4 x + 1 \right)} - \left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} x^{2} \]constant-multipleApply the constant multiple rule to 2*x**2.✓ Proved
- \[ = 2 x - \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} - \left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \frac{1}{2} \]derivative algebraDifferentiate the power terms. Simplify the expression inside the parentheses.✓ Proved
- \[ = 2 x - \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} + \frac{1}{2} - \frac{\left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \left(4 x + 1\right)}{4 x + 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = 2 x - \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} + \frac{1}{2} - \frac{4 \left(2 x^{2} + x + \frac{1}{8}\right)}{4 x + 1} \]derivativeDifferentiate the inner function 4*x + 1.✓ Proved
- \[ = 2 x - \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} + \frac{1}{2} - \frac{8 x^{2} + 4 x + \frac{1}{2}}{4 x + 1} \]algebra algebraMultiply the constant 4 into the numerator. Distribute the 4 in the numerator.✓ Proved
- \[ = 2 x - \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} + \frac{1}{2} - \frac{16 x^{2} + 8 x + 1}{8 x + 2} \]algebra algebra simplifyMultiply numerator and denominator by 2 to clear the fraction. Simplify the denominator. Final simplification of the expression.✓ Proved
Answer \( \left(- 4 x - 1\right) \ln{\left(4 x + 1 \right)} \)
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. How we verify
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 |
| 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 |
| 13 | ✓ 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 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 8*x + 2 = 0 |
| 15 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step. The final simplification steps are algebraically valid, and although the final expression does not immediately look like the stated answer, it is equivalent to it (the rational term simplifies to 2x + 1/2, which cancels the initial terms). No rule violations or logical errors are present.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies differentiation rules step-by-step. The final simplification steps are algebraically valid, and although the final expression does not immediately look like the stated answer, it is equivalent to it (the rational term simplifies to 2x + 1/2, which cancels the initial terms). No rule violations or logical errors are present.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — 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 the final result matches the stated answer.gpt-oss:20b: fail (error) 2026-10-16 — The final expression is not equivalent to the correct derivative; the simplification steps incorrectly cancel the remaining fraction term, resulting in an incomplete derivative.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The final answer provided in the steps (Step 15) does not match the stated answer. The steps fail to simplify the rational term (16*x**2 + 8*x + 1)/(2*(4*x + 1)) to 2*x + 1/2, which would cancel the leading terms 2*x + 1/2, leaving only the logarithmic term. As written, the solution is incomplete and incorrect.gpt-oss:20b: fail (error) 2026-10-16 — Step 15 incorrectly reverts the simplification performed in step 14, leaving the derivative in an inconsistent form. The final expression does not match the correct derivative, which should simplify to y = 2*x + 1/2 - (4*x+1)*log(4*x+1) - (8*x**2+4*x+1/2)/(4*x+1) and further simplifies to - (4*x+1)*log(4*x+1). The solution therefore contains a logical error in the final simplification step.
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 2026-10-19 with SymPy 1.14.0.