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 2.2282 · hard Video
Differentiate \( \displaystyle f(x) = - 5 x^{2} - \frac{5 x}{2} + \left(10 x^{2} + 5 x + \frac{5}{8}\right) \ln{\left(4 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- 5 x^{2} - \frac{5 x}{2} + \left(10 x^{2} + 5 x + \frac{5}{8}\right) \ln{\left(4 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{5 x}{2}\right) + \frac{d}{d x} \left(- 5 x^{2}\right) + \frac{d}{d x} \left(10 x^{2} + 5 x + \frac{5}{8}\right) \ln{\left(4 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(10 x^{2} + 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \ln{\left(4 x + 1 \right)} \frac{d}{d x} \left(10 x^{2} + 5 x + \frac{5}{8}\right) + \frac{d}{d x} \left(- \frac{5 x}{2}\right) + \frac{d}{d x} \left(- 5 x^{2}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(10 x^{2} + 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \left(\frac{d}{d x} \frac{5}{8} + \frac{d}{d x} 5 x + \frac{d}{d x} 10 x^{2}\right) \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- \frac{5 x}{2}\right) + \frac{d}{d x} \left(- 5 x^{2}\right) \]sumApply the sum rule to the derivative of the polynomial.✓ Proved
- \[ = \left(\frac{d}{d x} 5 x + \frac{d}{d x} 10 x^{2}\right) \ln{\left(4 x + 1 \right)} + \left(10 x^{2} + 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- \frac{5 x}{2}\right) + \frac{d}{d x} \left(- 5 x^{2}\right) \]constantThe derivative of the constant 5/8 is 0.✓ Proved
- \[ = \left(\frac{d}{d x} 10 x^{2} + 5\right) \ln{\left(4 x + 1 \right)} + \left(10 x^{2} + 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- \frac{5 x}{2}\right) + \frac{d}{d x} \left(- 5 x^{2}\right) \]powerDifferentiate 5*x.✓ Proved
- \[ = \left(20 x + 5\right) \ln{\left(4 x + 1 \right)} + \left(10 x^{2} + 5 x + \frac{5}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- \frac{5 x}{2}\right) + \frac{d}{d x} \left(- 5 x^{2}\right) \]powerDifferentiate 10*x**2.✓ Proved
- \[ = \left(20 x + 5\right) \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- \frac{5 x}{2}\right) + \frac{d}{d x} \left(- 5 x^{2}\right) + \frac{\left(10 x^{2} + 5 x + \frac{5}{8}\right) \frac{d}{d x} \left(4 x + 1\right)}{4 x + 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(20 x + 5\right) \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- \frac{5 x}{2}\right) + \frac{d}{d x} \left(- 5 x^{2}\right) + \frac{4 \left(10 x^{2} + 5 x + \frac{5}{8}\right)}{4 x + 1} \]powerDifferentiate the inner function 4*x + 1.✓ Proved
- \[ = - 10 x + \left(20 x + 5\right) \ln{\left(4 x + 1 \right)} - \frac{5}{2} + \frac{4 \left(10 x^{2} + 5 x + \frac{5}{8}\right)}{4 x + 1} \]derivativeEvaluate the remaining derivatives.✓ Proved
- \[ = - 10 x + \left(20 x + 5\right) \ln{\left(4 x + 1 \right)} - \frac{5}{2} + \frac{40 x^{2} + 20 x + \frac{5}{2}}{4 x + 1} \]algebraDistribute the 4 in the numerator.✓ 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 algebraMultiply numerator and denominator by 2 to clear the fraction 5/2. Distribute the 2 in the denominator.✓ Proved
- \[ = - 10 x + \left(20 x + 5\right) \ln{\left(4 x + 1 \right)} - \frac{5}{2} + \frac{40 x^{2} + 20 x + \frac{5}{2}}{4 x + 1} \]algebraDivide numerator and denominator by 2.✓ Proved
- \[ = - 10 x + \left(20 x + 5\right) \ln{\left(4 x + 1 \right)} - \frac{5}{2} + \frac{80 x^{2} + 40 x + 5}{8 x + 2} \]algebraRe-expand the denominator for clarity.✓ Proved
- \[ = - 10 x + \left(20 x + 5\right) \ln{\left(4 x + 1 \right)} - \frac{5}{2} + \frac{40 x^{2} + 20 x + \frac{5}{2}}{4 x + 1} \]simplifyFinal 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. 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 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 |
| 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 undefined where 4*x + 1 = 0 |
| 15 | ✓ 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 |
| 16 | ✓ 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: fail (style) — Step 6 labels the derivative of 5*x as "power", but the correct rule is "constant-multiple" (or simply "constant" for the derivative of a constant times x).qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step. The algebraic manipulations in the final steps are redundant but algebraically valid and do not constitute a defect under the given contract.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies differentiation rules step-by-step. The algebraic manipulations in the final steps are redundant but algebraically valid and do not constitute a defect under the given contract.gpt-oss:20b: fail (style) 2026-10-17 — Step 6 labels the derivative of 5*x as "power", but the correct rule is "constant-multiple" (or simply "constant" for the derivative of a constant times x).qwen3.6:27b-mlx: pass 2026-10-15 — 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, even if somewhat redundant.gpt-oss:20b: fail (style) 2026-10-15 — Steps 6 and 9 use the label "power" for differentiating linear terms (5*x and 4*x+1). The correct labels would be "constant-multiple" or "product" for 5*x and "constant-multiple" for 4*x+1. This mislabeling does not affect the algebraic correctness but violates the rule‑label granularity requirement.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The final answer provided in the 'Stated answer' field is incorrect; it omits the polynomial terms -10*x - 5/2 and the rational term resulting from the product rule, which are present in the final step of the derivation. The solution derives the correct expression but fails to match the stated answer.gpt-oss:20b: fail (style) 2026-10-14 — Steps 6 and 9 incorrectly label the derivative of a linear term and a linear inner function as "power"; the correct labels would be "constant-multiple" and "product" (or simply "derivative" for the inner function).
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 2026-10-17 with SymPy 1.14.0.