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 2.2130 · hard
Differentiate \( \displaystyle f(x) = \frac{3 x^{2}}{4} - \frac{x}{2} + \left(- \frac{3 x^{2}}{2} + x - \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} \).
- \[ \frac{d}{d x} \left(\frac{3 x^{2}}{4} - \frac{x}{2} + \left(- \frac{3 x^{2}}{2} + x - \frac{1}{6}\right) \ln{\left(3 x - 1 \right)}\right) \]Start 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
- \[ = \left(- \frac{3 x^{2}}{2} + x - \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \left(- \frac{d}{d x} \frac{1}{6} + \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right)\right) \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{3 x^{2}}{4} \]sumApply the sum rule to the first part of the product rule.✓ Proved
- \[ = \left(\frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + 1\right) \ln{\left(3 x - 1 \right)} + \left(- \frac{3 x^{2}}{2} + x - \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{3 x^{2}}{4} \]constant algebraDifferentiate the constant terms. Simplify the expression inside the parentheses.✓ Proved
- \[ = \left(\frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + 1\right) \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{3 x^{2}}{4} + \frac{\left(- \frac{3 x^{2}}{2} + x - \frac{1}{6}\right) \frac{d}{d x} \left(3 x - 1\right)}{3 x - 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(\frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + 1\right) \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{3 x^{2}}{4} + \frac{3 \left(- \frac{3 x^{2}}{2} + x - \frac{1}{6}\right)}{3 x - 1} \]chainDifferentiate the inner function of the logarithm.✓ Proved
- \[ = \left(\frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + 1\right) \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{3 x^{2}}{4} + \frac{- \frac{9 x^{2}}{2} + 3 x - \frac{1}{2}}{3 x - 1} \]algebraSimplify the product.✓ Proved
- \[ = \left(\frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + 1\right) \ln{\left(3 x - 1 \right)} - \frac{\frac{d}{d x} x}{2} + \frac{3 \frac{d}{d x} x^{2}}{4} + \frac{- \frac{9 x^{2}}{2} + 3 x - \frac{1}{2}}{3 x - 1} \]constant-multipleFactor out the constants.✓ Proved
- \[ = \frac{3 x}{2} + \left(\frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + 1\right) \ln{\left(3 x - 1 \right)} - \frac{1}{2} + \frac{- \frac{9 x^{2}}{2} + 3 x - \frac{1}{2}}{3 x - 1} \]power algebraDifferentiate the power terms. Simplify the first term.✓ Proved
- \[ = \frac{3 x}{2} + \left(1 - 3 x\right) \ln{\left(3 x - 1 \right)} - \frac{1}{2} + \frac{- \frac{9 x^{2}}{2} + 3 x - \frac{1}{2}}{3 x - 1} \]power algebra constant simplifyDifferentiate the remaining power term. Distribute the 3 in the last term. The derivative of the constant is zero. Final simplification.✓ Proved
Answer \( \left(1 - 3 x\right) \ln{\left(3 x - 1 \right)} \)
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 |
| 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 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 |
| 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 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 16 | ✓ 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) — Step 13 incorrectly labels the operation as "power"; it merely substitutes the already‑computed derivative of −3*x**2/2, which is an algebraic simplification, not a power rule application.qwen3.6:27b-mlx: fail (error) — Step 15 introduces a spurious '+ 0' term with the label 'constant', which is not a valid differentiation step and does not appear in the previous line. Step 16 then removes it, making the entire addition and removal of zero a nonsensical artifact.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-12 — Step 13 incorrectly labels the operation as "power"; it merely substitutes the already‑computed derivative of −3*x**2/2, which is an algebraic simplification, not a power rule application.qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 15 introduces a spurious '+ 0' term with the label 'constant', which is not a valid differentiation step and does not appear in the previous line. Step 16 then removes it, making the entire addition and removal of zero a nonsensical artifact.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 15 introduces a '+ 0' term with the label 'constant', but the previous step (14) did not contain this term, and no differentiation rule was applied to generate it. This is an unjustified addition of a term, violating the requirement that steps be algebraically equal transformations of the previous line.
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.