Derivative of \( \displaystyle - \frac{5 x^{2}}{4} - x + \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \)
Problem 2.2376 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{4} - x + \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(- \frac{5 x^{2}}{4} - x + \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) \]productApply the product rule to the last term.✓ Proved
- \[ = \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} + \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) \]derivativeDifferentiate the polynomial part of the product.✓ Proved
- \[ = \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) + \frac{\left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \frac{d}{d x} \left(5 x + 2\right)}{5 x + 2} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) + \frac{5 \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right)}{5 x + 2} \]derivative algebraDifferentiate the inner function 5*x + 2. Simplify the product of the constant and the fraction.✓ Proved
- \[ = \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{5 x^{2}}{4}\right) + \frac{\frac{25 x^{2}}{2} + 10 x + 2}{5 x + 2} \]algebraDistribute the 5 into the numerator.✓ Proved
- \[ = - \frac{5 x}{2} + \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} - 1 + \frac{\frac{25 x^{2}}{2} + 10 x + 2}{5 x + 2} \]derivativeEvaluate the remaining derivatives.✓ Proved
- \[ = - \frac{5 x}{2} + \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} - 1 + \frac{25 x^{2} + 20 x + 4}{10 x + 4} \]algebra algebra simplifyFind a common denominator for the fraction term. Expand the denominator. Final simplification of the expression.✓ Proved
Answer \( \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} \)
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 undefined where 5*x + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 10*x + 4 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 10*x + 4 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 10*x + 4 = 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) — The final expression still contains canceling terms that were not simplified away; a simplification step should combine -5*x/2 - 1 + (25*x**2+20*x+4)/(10*x+4) to zero, yielding the correct result (5*x+2)*log(5*x+2).qwen3.6:27b-mlx: fail (misleading) — The solution fails to simplify the rational term (25*x**2 + 20*x + 4)/(10*x + 4) to (5*x + 2)/2, which is necessary to cancel the linear terms -5*x/2 - 1 and arrive at the stated answer. As presented, the final expression does not equal the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The solution fails to simplify the rational term (25*x**2 + 20*x + 4)/(10*x + 4) to (5*x + 2)/2, which is necessary to cancel the linear terms -5*x/2 - 1 and arrive at the stated answer. As presented, the final expression does not equal the stated answer.gpt-oss:20b: fail (style) 2026-10-18 — The final expression still contains canceling terms that were not simplified away; a simplification step should combine -5*x/2 - 1 + (25*x**2+20*x+4)/(10*x+4) to zero, yielding the correct result (5*x+2)*log(5*x+2).qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to simplify the rational term (25*x**2 + 20*x + 4)/(10*x + 4) to (5*x + 2)/2, which is necessary to cancel the linear terms -5*x/2 - 1 and arrive at the stated answer (5*x + 2)*log(5*x + 2). The final expression in step 12 is not equal to the stated answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to simplify the rational term (25*x**2 + 20*x + 4)/(10*x + 4) to 5*x/2 + 1, which is necessary to cancel the -5*x/2 - 1 terms and reach the stated answer. The final expression in step 12 is algebraically unequal to the stated answer.gpt-oss:20b: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution never simplifies the fraction \n\\((25x^2/2+10x+2)/(5x+2)\\) to cancel with the terms \\(-5x/2-1\\). As a result the final expression contai
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 2026-10-18 with SymPy 1.14.0.