Derivative of \( \displaystyle x^{2} - \frac{3 x}{2} + \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \ln{\left(4 x - 3 \right)} \)
Problem 2.2426 · hard Video
Differentiate \( \displaystyle f(x) = x^{2} - \frac{3 x}{2} + \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \ln{\left(4 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(x^{2} - \frac{3 x}{2} + \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \ln{\left(4 x - 3 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} x^{2} + \frac{d}{d x} \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \ln{\left(4 x - 3 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} x^{2} \]productApply the product rule to the third term.✓ Proved
- \[ = \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \left(- \frac{d}{d x} \frac{9}{8} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 2 x^{2}\right)\right) \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} x^{2} \]sumApply the sum rule to the polynomial term.✓ Proved
- \[ = \left(3 \frac{d}{d x} x - 2 \frac{d}{d x} x^{2}\right) \ln{\left(4 x - 3 \right)} + \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} x^{2} \]constant constantDifferentiate the constant term. Differentiate the constant term.✓ Proved
- \[ = \left(3 - 2 \frac{d}{d x} x^{2}\right) \ln{\left(4 x - 3 \right)} + \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} x^{2} \]derivativeDifferentiate the x terms.✓ Proved
- \[ = \left(3 - 2 \frac{d}{d x} x^{2}\right) \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} x^{2} + \frac{\left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(3 - 2 \frac{d}{d x} x^{2}\right) \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} x^{2} + \frac{4 \left(- 2 x^{2} + 3 x - \frac{9}{8}\right)}{4 x - 3} \]derivativeDifferentiate the inner function of the logarithm.✓ Proved
- \[ = \left(3 - 2 \frac{d}{d x} x^{2}\right) \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} x^{2} + \frac{- 8 x^{2} + 12 x - \frac{9}{2}}{4 x - 3} \]algebraSimplify the product of the constant and the fraction.✓ Proved
- \[ = 2 x + \left(3 - 4 x\right) \ln{\left(4 x - 3 \right)} - \frac{3}{2} + \frac{- 8 x^{2} + 12 x - \frac{9}{2}}{4 x - 3} \]derivative algebra algebraDifferentiate the remaining power terms. Simplify the polynomial term. Distribute the 4 in the last term.✓ Proved
- \[ = 2 x + \left(3 - 4 x\right) \ln{\left(4 x - 3 \right)} - \frac{3}{2} + \frac{- 16 x^{2} + 24 x - 9}{8 x - 6} \]algebra algebra simplifySimplify the fraction by combining terms. Expand the denominator. Final simplification.✓ Proved
Answer \( \left(3 - 4 x\right) \ln{\left(4 x - 3 \right)} \)
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 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 undefined where 8*x - 6 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 8*x - 6 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 8*x - 6 = 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 solution never simplifies the rational term (-16*x**2+24*x-9)/(2*(4*x-3)) to (3-4*x)/2, so the final expression does not reduce to the correct derivative (3-4*x)*log(4*x-3).qwen3.6:27b-mlx: fail (error) — The solution fails to simplify the rational term (-16*x**2 + 24*x - 9)/(2*(4*x - 3)) to 4*x - 3, which is required to reach the stated answer. Consequently, the final expression does not match the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to simplify the rational term (-16*x**2 + 24*x - 9)/(2*(4*x - 3)) to 4*x - 3, which is required to reach the stated answer. Consequently, the final expression does not match the stated answer.gpt-oss:20b: fail (error) 2026-10-19 — The solution never simplifies the rational term (-16*x**2+24*x-9)/(2*(4*x-3)) to (3-4*x)/2, so the final expression does not reduce to the correct derivative (3-4*x)*log(4*x-3).qwen3.6:27b-mlx: fail (style) 2026-10-16 — Steps 5 and 6 are redundant; Step 5 applies the constant rule to the term 9/8, and Step 6 repeats this action on the resulting zero, violating the 'one change per step' principle. Additionally, Step 16 reverts the simplification from Step 14, making the final line algebraically identical to Step 14 but with a misleading note about 'final simplification'.gpt-oss:20b: fail (error) 2026-10-16 — The final derivative contains an extra rational term that should cancel out; the solution fails to simplify the expression to the correct result (3-4*x)*log(4*x-3).qwen3.6:27b-mlx: fail (style) 2026-10-16 — Steps 5 and 6 are redundant; Step 5 applies the constant rule to the term 9/8, and Step 6 repeats the same operation on the resulting zero, violating the one-change-per-step principle. Additionally, the solution fails to simplify the rational term to its constant value (-2), leaving the answer in a non-canonical form compared to the stated answer.gpt-oss:20b: fail (error) 2026-10-16 — The final expression contains extra terms that should cancel, so the solution does not correctly simplify to the stated answer (3-4*x)*log(4*x-3).
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.