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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 2.2440 · hard Video

Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{4} + \frac{x}{2} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} \).
  1. \[ \frac{d}{d x} \left(- \frac{3 x^{2}}{4} + \frac{x}{2} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)}\right) \]
    derivativeDifferentiate the entire function.✓ Proved
  2. \[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} \]
    sum constant-multipleApply the sum rule. Factor out the constants.✓ Proved
  3. \[ = \frac{\frac{d}{d x} x}{2} - \frac{3 \frac{d}{d x} x^{2}}{4} + \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} \]
    constant-multipleFactor out the constants from each term.✓ Proved
  4. \[ = - \frac{3 x}{2} + \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} \]
    powerDifferentiate the power terms.✓ Proved
  5. \[ = - \frac{3 x}{2} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \ln{\left(3 x - 1 \right)} \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) + \frac{1}{2} \]
    productApply the product rule to the third term.✓ Proved
  6. \[ = - \frac{3 x}{2} + \left(- \frac{d}{d x} x + \frac{3 \frac{d}{d x} x^{2}}{2}\right) \ln{\left(3 x - 1 \right)} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \frac{1}{2} \]
    sumDifferentiate the polynomial inside the product rule.✓ Proved
  7. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \frac{1}{2} \]
    power algebraDifferentiate the power terms in the first part of the product rule. Simplify the expression inside the parentheses.✓ Proved
  8. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} + \frac{\left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \frac{d}{d x} \left(3 x - 1\right)}{3 x - 1} \]
    chainApply the chain rule to the logarithm.✓ Proved
  9. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} + \frac{3 \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right)}{3 x - 1} \]
    constant-multipleDifferentiate the inner function 3x - 1.✓ Proved
  10. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} + \frac{\frac{9 x^{2}}{2} - 3 x + \frac{1}{2}}{3 x - 1} \]
    algebra simplify algebraMultiply the 3 into the numerator. Simplify the first two terms. Factor out 1/2 from the numerator.✓ Proved
  11. \[ = \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} \]
    algebra simplify algebra simplifyRecognize the perfect square trinomial. Cancel the common factor (3x - 1). Distribute the 1/2. Combine all terms to find the final simplified derivative.✓ Proved
Answer \( \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} \)
Top: the graph of f = -3x²/4 + x/2 + (3x²/2 - x + 1/6)·log(3x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3x/2 + (3x - 1)·log(3x - 1) + 1/2 + 3(3x²/2 - x + 1/6)/(3x - 1).

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify

The full receipt
LineStatusChecked byDetail
1✓ Provedsympy 1.14.0line 1 is the problem as stated
2✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
3✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
4✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
5✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
6✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
7✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
8✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
9✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
10✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 3*x - 1 = 0
11✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 3*x - 1 = 0
12✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 3*x - 1 = 0
13✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 3*x - 1 = 0
14✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 3*x - 1 = 0
15✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 3*x - 1 = 0
16✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
17✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
18✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
answer✓ Provedsympy 1.14.0final line against the stated answer: simplify(a - b) reduced to 0
log is undefined for non-positive arguments
answer, a second way✓ Provedsympy 1.14.0SymPy differentiated f directly and got the stated answer

Reviewers

  • gpt-oss:20b: pass
  • qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
  • qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
  • gpt-oss:20b: pass 2026-10-19
  • qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. All labels are appropriate for the operations performed, and the algebraic simplifications are valid.
  • gpt-oss:20b: pass 2026-10-17
  • qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
  • gpt-oss:20b: pass 2026-10-17

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.

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