Derivative of \( \displaystyle - \frac{5 x^{2}}{2} - x + \left(5 x^{2} + 2 x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)} \)
Problem 2.1515 · hard
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{2} - x + \left(5 x^{2} + 2 x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- \frac{5 x^{2}}{2} - x + \left(5 x^{2} + 2 x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) + \frac{d}{d x} \left(5 x^{2} + 2 x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(5 x^{2} + 2 x + \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} \left(5 x^{2} + 2 x + \frac{1}{5}\right) - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(10 x + 2\right) \ln{\left(5 x + 1 \right)} + \left(5 x^{2} + 2 x + \frac{1}{5}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) \]derivativeDifferentiate the polynomial part.✓ Proved
- \[ = \left(10 x + 2\right) \ln{\left(5 x + 1 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) + \frac{\left(5 x^{2} + 2 x + \frac{1}{5}\right) \frac{d}{d x} \left(5 x + 1\right)}{5 x + 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(10 x + 2\right) \ln{\left(5 x + 1 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) + \frac{5 \left(5 x^{2} + 2 x + \frac{1}{5}\right)}{5 x + 1} \]derivativeDifferentiate the inner function of the logarithm.✓ Proved
- \[ = \left(10 x + 2\right) \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \left(- \frac{5 x^{2}}{2}\right) - 1 + \frac{5 \left(5 x^{2} + 2 x + \frac{1}{5}\right)}{5 x + 1} \]derivativeDifferentiate the x term.✓ Proved
- \[ = - 5 x + \left(10 x + 2\right) \ln{\left(5 x + 1 \right)} - 1 + \frac{5 \left(5 x^{2} + 2 x + \frac{1}{5}\right)}{5 x + 1} \]derivativeDifferentiate the first term.✓ Proved
- \[ = - 5 x + \left(10 x + 2\right) \ln{\left(5 x + 1 \right)} - 1 + \frac{25 x^{2} + 10 x + 1}{5 x + 1} \]algebraDistribute the 5 in the last term.✓ Proved
- \[ = \left(10 x + 2\right) \ln{\left(5 x + 1 \right)} \]algebra simplify simplifyRecognize the perfect square trinomial. Simplify the fraction. Combine like terms.✓ Proved
Answer \( 2 \left(5 x + 1\right) \ln{\left(5 x + 1 \right)} \)
✓ 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 5*x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 12 | ✓ 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 simplification at the end is correct and leads to the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification at the end is correct and leads to the stated answer.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.