Derivative of \( \displaystyle \frac{3 x^{2}}{4} + \frac{x}{2} - \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \ln{\left(3 x + 1 \right)} \)
Problem 2.2534 · hard Video
Differentiate \( \displaystyle f(x) = \frac{3 x^{2}}{4} + \frac{x}{2} - \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \ln{\left(3 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(\frac{3 x^{2}}{4} + \frac{x}{2} - \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \ln{\left(3 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{3 x^{2}}{4} - \frac{d}{d x} \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \ln{\left(3 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = - \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} - \ln{\left(3 x + 1 \right)} \frac{d}{d x} \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{3 x^{2}}{4} \]productApply the product rule to the third term.✓ Proved
- \[ = - \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} - \left(\frac{d}{d x} \frac{1}{6} + \frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{2}\right) \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{3 x^{2}}{4} \]sumApply the sum rule to the first part of the product rule.✓ Proved
- \[ = - \left(\frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{2}\right) \ln{\left(3 x + 1 \right)} - \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{3 x^{2}}{4} \]constant sumThe derivative of the constant 1/6 is 0. Apply the sum rule again.✓ Proved
- \[ = \frac{3 x}{2} - \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} - \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \frac{1}{2} \]powerDifferentiate the power terms.✓ Proved
- \[ = \frac{3 x}{2} - \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} + \frac{1}{2} - \frac{\left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{3 x}{2} - \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} + \frac{1}{2} - \frac{3 \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right)}{3 x + 1} \]derivativeDifferentiate the inner function 3x + 1.✓ Proved
- \[ = \frac{3 x}{2} - \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} + \frac{1}{2} - \frac{\frac{9 x^{2}}{2} + 3 x + \frac{1}{2}}{3 x + 1} \]algebra simplify simplifyDistribute the 3 into the fraction. Final simplification. Clean up the expression.✓ Proved
Answer \( \left(- 3 x - 1\right) \ln{\left(3 x + 1 \right)} \)
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 3*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*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) — The solution stops short of simplifying the final expression to the expected form (-3*x-1)*log(3*x+1).qwen3.6:27b-mlx: fail (error) — The solution fails to simplify the final rational term (9*x**2/2 + 3*x + 1/2)/(3*x + 1) to 3*x/2 + 1/2, which would cancel the initial terms to yield the stated answer. As written, the final line is not algebraically equal to the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-01 — The solution fails to simplify the final rational term (9*x**2/2 + 3*x + 1/2)/(3*x + 1) to 3*x/2 + 1/2, which would cancel the initial terms to yield the stated answer. As written, the final line is not algebraically equal to the stated answer.gpt-oss:20b: fail (style) 2026-11-01 — The solution stops short of simplifying the final expression to the expected form (-3*x-1)*log(3*x+1).qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to simplify the rational term (9*x**2/2 + 3*x + 1/2)/(3*x + 1) to (3*x + 1)/2, which is necessary to cancel the linear terms and arrive at the stated answer. Consequently, the final expression in step 12 does not equal the stated answer.gpt-oss:20b: fail (error) 2026-10-31 — The final expression does not simplify to the expected derivative (-3*x-1)*log(3*x+1); the algebraic simplification step is missing, leading to an incorrect final answer.qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to simplify the rational term (9*x**2/2 + 3*x + 1/2)/(3*x + 1) to (3*x + 1)/2, which is necessary to cancel with the linear terms and reach the stated answer. Consequently, the final expression in step 12 does not equal the stated answer.gpt-oss:20b: pass 2026-10-31
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 2026-11-01 with SymPy 1.14.0.