Derivative of \( \displaystyle - 2 x^{2} - x + \left(4 x^{2} + 2 x + \frac{1}{4}\right) \ln{\left(4 x + 1 \right)} \)
Problem 2.2128 · hard
Differentiate \( \displaystyle f(x) = - 2 x^{2} - x + \left(4 x^{2} + 2 x + \frac{1}{4}\right) \ln{\left(4 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- 2 x^{2} - x + \left(4 x^{2} + 2 x + \frac{1}{4}\right) \ln{\left(4 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{d}{d x} \left(4 x^{2} + 2 x + \frac{1}{4}\right) \ln{\left(4 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(4 x^{2} + 2 x + \frac{1}{4}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \ln{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x^{2} + 2 x + \frac{1}{4}\right) + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- 2 x^{2}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(8 x + 2\right) \ln{\left(4 x + 1 \right)} + \left(4 x^{2} + 2 x + \frac{1}{4}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- 2 x^{2}\right) \]derivativeDifferentiate the polynomial term.✓ Proved
- \[ = \left(8 x + 2\right) \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{\left(4 x^{2} + 2 x + \frac{1}{4}\right) \frac{d}{d x} \left(4 x + 1\right)}{4 x + 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(8 x + 2\right) \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{4 \left(4 x^{2} + 2 x + \frac{1}{4}\right)}{4 x + 1} \]derivativeDifferentiate the inner function 4*x + 1.✓ Proved
- \[ = \left(8 x + 2\right) \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{16 x^{2} + 8 x + 1}{4 x + 1} \]algebraMultiply the numerator by 4.✓ Proved
- \[ = 4 x + \left(8 x + 2\right) \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- 2 x^{2}\right) + 1 \]algebra simplifyRecognize the perfect square trinomial. Simplify the fraction.✓ Proved
- \[ = \left(8 x + 2\right) \ln{\left(4 x + 1 \right)} \]derivative simplifyDifferentiate the remaining polynomial terms. Combine like terms.✓ Proved
Answer \( 2 \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} \)
✓ 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ 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 step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and final result are correct.
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, adhering to the one-change-per-step constraint. The algebraic simplifications and final result are correct.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and final result are correct.
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.