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 2.2112 · hard
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
- \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \ln{\left(3 x + 1 \right)} \]sumApply the sum rule to separate the terms.✓ Proved
- \[ = \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} \frac{d}{d x} \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} + \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) \]derivativeDifferentiate the polynomial part of the product.✓ Proved
- \[ = \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{\left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{3 \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right)}{3 x + 1} \]derivativeDifferentiate the inner function 3x + 1.✓ Proved
- \[ = \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{\frac{9 x^{2}}{2} + 3 x + \frac{1}{2}}{3 x + 1} \]algebraDistribute the 3 and simplify the fraction.✓ Proved
- \[ = - \frac{3 x}{2} + \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} - \frac{1}{2} + \frac{\frac{9 x^{2}}{2} + 3 x + \frac{1}{2}}{3 x + 1} \]derivative constant simplifyDifferentiate the power and linear terms. Note that the constant term is zero. Final simplification of 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.
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 3*x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 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 |
| 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: fail (misleading) — Step 9 and Step 10 are redundant and misleading; Step 9 adds a zero (which is not a differentiation rule, but even if labeled 'algebra' or 'simplify', it changes nothing) and Step 10 removes it. More critically, the solution fails to actually simplify the rational term `(9*x**2/2 + 3*x + 1/2)/(3*x + 1)` to `3*x/2 + 1/2`, which would cancel the `-3*x/2 - 1/2` terms to yield the stated answer. The final expression in Step 10 is algebraically equal to the stated answer, but the steps shown do not perform the necessary simplification to reach that form, leaving the student with an unsimplified expression that looks different from the target. The label 'simplify' in Step 10 is used to hide the lack of actual simplification work.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (misleading) 2026-10-12 — Step 9 and Step 10 are redundant and misleading; Step 9 adds a zero (which is not a differentiation rule, but even if labeled 'algebra' or 'simplify', it changes nothing) and Step 10 removes it. More critically, the solution fails to actually simplify the rational term `(9*x**2/2 + 3*x + 1/2)/(3*x + 1)` to `3*x/2 + 1/2`, which would cancel the `-3*x/2 - 1/2` terms to yield the stated answer. The final expression in Step 10 is algebraically equal to the stated answer, but the steps shown do not perform the necessary simplification to reach that form, leaving the student with an unsimplified expression that looks different from the target. The label 'simplify' in Step 10 is used to hide the lack of actual simplification work.gpt-oss:20b: fail (misleading) 2026-10-12 — The solution fails to cancel the terms -3*x/2 - 1/2 with the fraction (9*x**2/2 + 3*x + 1/2)/(3*x + 1). The fraction actually simplifies to (3/2)*x + 1/2, which cancels the preceding linear terms, leaving the correct derivative (3*x + 1)*log(3*x + 1).qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 7 claims to simplify the fraction (9*x**2/2 + 3*x + 1/2)/(3*x + 1) via algebra, but this fraction does not simplify to a constant or a simpler term that cancels with the previous terms to yield the stated answer. The stated answer is mathematically incorrect; the derivative of the given function is not (3*x + 1)*log(3*x + 1). The solution fails to correctly simplify the remaining terms, leading to a wrong final result.
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.