Derivative of \( \displaystyle - \frac{3 x^{2}}{4} - x + \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \)
Problem 2.2414 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{4} - x + \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(- \frac{3 x^{2}}{4} - x + \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)}\right) \]derivativeStart with the derivative of the entire function.✓ Proved
- \[ = - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \]sumApply the sum rule to separate the terms.✓ Proved
- \[ = \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \left(\frac{d}{d x} \frac{2}{3} + \frac{d}{d x} 2 x + \frac{d}{d x} \frac{3 x^{2}}{2}\right) \ln{\left(3 x + 2 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) \]sumApply the sum rule to the first part of the product.✓ Proved
- \[ = \left(\frac{d}{d x} 2 x + \frac{d}{d x} \frac{3 x^{2}}{2}\right) \ln{\left(3 x + 2 \right)} + \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) \]constant algebraThe derivative of the constant 2/3 is zero. Simplify the expression by removing the zero term.✓ Proved
- \[ = \left(\frac{d}{d x} \frac{3 x^{2}}{2} + 2\right) \ln{\left(3 x + 2 \right)} + \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) - 1 \]constantEvaluate the derivatives of the simple terms.✓ Proved
- \[ = \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} + \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) - 1 \]powerApply the power rule to the term 3*x**2/2.✓ Proved
- \[ = \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) - 1 + \frac{\left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} \]chainApply the chain rule to the log term.✓ Proved
- \[ = \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) - 1 + \frac{3 \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right)}{3 x + 2} \]derivativeEvaluate the derivative of the inner function 3*x + 2.✓ Proved
- \[ = - \frac{3 x}{2} + \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} - 1 + \frac{3 \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right)}{3 x + 2} \]powerEvaluate the derivative of -3*x**2/4.✓ Proved
- \[ = - \frac{3 x}{2} + \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} - 1 + \frac{\frac{9 x^{2}}{2} + 6 x + 2}{3 x + 2} \]algebra simplify simplifyDistribute the 3 into the polynomial term. The expression is already in its simplest form. Final simplified result.✓ Proved
Answer \( \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} \)
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 |
| 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 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 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 (error) — Step 13 incorrectly adds and subtracts extra terms, resulting in an algebraic error. The expression should not contain the extra "-1 + 1" pair.qwen3.6:27b-mlx: fail (style) — Step 7 applies the 'constant' rule to evaluate Derivative(x, x) and Derivative(2*x, x), but the derivative of x is a basic power rule application, and the derivative of 2x involves the constant-multiple rule; labeling these as 'constant' is incorrect. Additionally, Step 13 and 14 are redundant no-ops that add no value.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-19 — Step 7 applies the 'constant' rule to evaluate Derivative(x, x) and Derivative(2*x, x), but the derivative of x is a basic power rule application, and the derivative of 2x involves the constant-multiple rule; labeling these as 'constant' is incorrect. Additionally, Step 13 and 14 are redundant no-ops that add no value.gpt-oss:20b: fail (error) 2026-10-19 — Step 13 incorrectly adds and subtracts extra terms, resulting in an algebraic error. The expression should not contain the extra "-1 + 1" pair.qwen3.6:27b-mlx: fail (style) 2026-10-16 — Step 7 applies the derivative rule to multiple terms simultaneously (Derivative(x, x) and Derivative(2*x, x)), violating the one-change-per-step constraint. Steps 13 and 14 are redundant and do not perform any actual simplification.gpt-oss:20b: fail (style) 2026-10-16 — Step 13 applies two changes (adding -1 and +1) in one step and is labeled "simplify". Each step must change only one thing; this violates the granularity rule.qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 7 applies the 'constant' rule but performs multiple distinct operations: it evaluates Derivative(x, x) to 1, Derivative(2*x, x) to 2, and removes the zero term from the previous step. This violates the one-rule-per-step constraint. Additionally, Step 13 adds and subtracts 1 with no mathematical purpose, which is misleading and violates the single-change rule.gpt-oss:20b: fail (error) 2026-10-16 — Step 13 adds an unnecessary "-1 + 1" pair, applying two operations (adding terms and simplifying) in one step, which violates the single‑rule‑per‑step rule and introduces an incorrect intermediate expression.
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 2026-10-19 with SymPy 1.14.0.