Derivative of \( \displaystyle - x^{2} - \frac{x}{2} + \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)} \)
Problem 2.2182 · hard
Differentiate \( \displaystyle f(x) = - x^{2} - \frac{x}{2} + \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- x^{2} - \frac{x}{2} + \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - 2 x + \frac{d}{d x} \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)} - \frac{1}{2} \]sumDifferentiate the terms separately.✓ Proved
- \[ = - 2 x + \left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \ln{\left(4 x + 1 \right)} \frac{d}{d x} \left(2 x^{2} + x + \frac{1}{8}\right) - \frac{1}{2} \]productApply the product rule to the second term.✓ Proved
- \[ = - 2 x + \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} + \left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} - \frac{1}{2} \]derivativeDifferentiate the polynomial factor.✓ 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 of the logarithm.✓ 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 algebra algebra algebra algebra simplify simplifySimplify the expression. Distribute the 4 into the numerator. Add and subtract 1/2 to facilitate simplification. Combine terms over a common denominator. Expand the numerator. Simplify the fraction. Remove zero.✓ Proved
- \[ = \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} \]algebra algebra simplifyPerform polynomial division on the fraction. Combine the linear terms. Final simplification.✓ Proved
Answer \( \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} \)
✓ 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 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 |
| 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 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments 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 |
| 16 | ✓ 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 and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied.
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 2026-10-13 with SymPy 1.14.0.