Derivative of \( \displaystyle - 3 x^{2} + \frac{9 x}{2} + \left(6 x^{2} - 9 x + \frac{27}{8}\right) \ln{\left(4 x - 3 \right)} \)
Problem 2.2400 · hard Video
Differentiate \( \displaystyle f(x) = - 3 x^{2} + \frac{9 x}{2} + \left(6 x^{2} - 9 x + \frac{27}{8}\right) \ln{\left(4 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(- 3 x^{2} + \frac{9 x}{2} + \left(6 x^{2} - 9 x + \frac{27}{8}\right) \ln{\left(4 x - 3 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{9 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) + \frac{d}{d x} \left(6 x^{2} - 9 x + \frac{27}{8}\right) \ln{\left(4 x - 3 \right)} \]sumApply the sum rule to separate the terms.✓ Proved
- \[ = \left(6 x^{2} - 9 x + \frac{27}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} \left(6 x^{2} - 9 x + \frac{27}{8}\right) + \frac{d}{d x} \frac{9 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(6 x^{2} - 9 x + \frac{27}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \left(\frac{d}{d x} \frac{27}{8} - \frac{d}{d x} 9 x + \frac{d}{d x} 6 x^{2}\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \frac{9 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) \]sumApply the sum rule to the polynomial part.✓ Proved
- \[ = \left(- \frac{d}{d x} 9 x + \frac{d}{d x} 6 x^{2}\right) \ln{\left(4 x - 3 \right)} + \left(6 x^{2} - 9 x + \frac{27}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \frac{9 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) \]constant algebraThe derivative of the constant 27/8 is 0. Simplify the expression inside the parentheses.✓ Proved
- \[ = \left(12 x - 9\right) \ln{\left(4 x - 3 \right)} + \left(6 x^{2} - 9 x + \frac{27}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \frac{9 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) \]powerDifferentiate the polynomial terms.✓ Proved
- \[ = \left(12 x - 9\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \frac{9 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) + \frac{\left(6 x^{2} - 9 x + \frac{27}{8}\right) \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(12 x - 9\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \frac{9 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) + \frac{4 \left(6 x^{2} - 9 x + \frac{27}{8}\right)}{4 x - 3} \]derivativeDifferentiate the inner function 4*x - 3.✓ Proved
- \[ = - 6 x + \left(12 x - 9\right) \ln{\left(4 x - 3 \right)} + \frac{9}{2} + \frac{4 \left(6 x^{2} - 9 x + \frac{27}{8}\right)}{4 x - 3} \]derivativeEvaluate the remaining derivatives.✓ Proved
- \[ = - 6 x + \left(12 x - 9\right) \ln{\left(4 x - 3 \right)} + \frac{9}{2} + \frac{24 x^{2} - 36 x + \frac{27}{2}}{4 x - 3} \]algebraDistribute the 4 into the numerator.✓ Proved
- \[ = - 6 x + \left(12 x - 9\right) \ln{\left(4 x - 3 \right)} + \frac{9}{2} + \frac{24 x^{2} - 36 x + 13.5}{4 x - 3} \]simplifyConvert the fraction to decimal form for final simplification.✓ Proved
Answer \( 3 \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} \)
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 4*x - 3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 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) — The final expression does not equal the claimed answer 3*(4*x-3)*log(4*x-3). The derivative was not simplified correctly, and the solution incorrectly concludes with a different result.qwen3.6:27b-mlx: fail (style) — Step 7 applies the power rule to multiple terms simultaneously, violating the one-change-per-step constraint. Step 12 introduces a decimal representation which is mathematically imprecise and stylistically poor for an exact calculus solution.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-18 — Step 7 applies the power rule to multiple terms simultaneously, violating the one-change-per-step constraint. Step 12 introduces a decimal representation which is mathematically imprecise and stylistically poor for an exact calculus solution.gpt-oss:20b: fail (error) 2026-10-18 — The final expression does not equal the claimed answer 3*(4*x-3)*log(4*x-3). The derivative was not simplified correctly, and the solution incorrectly concludes with a different result.qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 7 applies the power rule to differentiate two terms (6*x**2 and 9*x) simultaneously, violating the one-change-per-step constraint. Additionally, Step 10 applies the derivative rule to two separate terms (-3*x**2 and 9*x/2) in a single step.gpt-oss:20b: fail (error) 2026-10-16 — The final derivative expression is incorrect; the correct derivative is not 3*(4*x-3)*log(4*x-3).qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 10 applies the 'derivative' rule to two distinct terms (-3*x**2 and 9*x/2) simultaneously, violating the one-change-per-step constraint. Additionally, Step 12 introduces a decimal approximation (13.5) which is mathematically imprecise and stylistically poor for an exact calculus solution.gpt-oss:20b: pass 2026-10-15
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.