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Derivative of \( \displaystyle - x^{2} + x + \left(2 x^{2} - 2 x + \frac{1}{2}\right) \ln{\left(2 x - 1 \right)} \)

Problem 2.2383 · hard Video

Differentiate \( \displaystyle f(x) = - x^{2} + x + \left(2 x^{2} - 2 x + \frac{1}{2}\right) \ln{\left(2 x - 1 \right)} \).
  1. \[ \frac{d}{d x} \left(- x^{2} + x + \left(2 x^{2} - 2 x + \frac{1}{2}\right) \ln{\left(2 x - 1 \right)}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} \left(2 x^{2} - 2 x + \frac{1}{2}\right) \ln{\left(2 x - 1 \right)} \]
    sumApply the sum rule.✓ Proved
  3. \[ = \left(2 x^{2} - 2 x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x^{2} - 2 x + \frac{1}{2}\right) + \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) \]
    productApply the product rule to the third term.✓ Proved
  4. \[ = \left(2 x^{2} - 2 x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \left(\frac{d}{d x} \frac{1}{2} - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x^{2}\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) \]
    sumApply the sum rule to the polynomial term.✓ Proved
  5. \[ = \left(- 2 \frac{d}{d x} x + \frac{d}{d x} 2 x^{2}\right) \ln{\left(2 x - 1 \right)} + \left(2 x^{2} - 2 x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) \]
    constantDifferentiate the constant term.✓ Proved
  6. \[ = \left(- 2 \frac{d}{d x} x + 2 \frac{d}{d x} x^{2}\right) \ln{\left(2 x - 1 \right)} + \left(2 x^{2} - 2 x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) \]
    constant-multipleFactor out the constant 2.✓ Proved
  7. \[ = 4 x \ln{\left(2 x - 1 \right)} + \left(2 x^{2} - 2 x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} - 2 \ln{\left(2 x - 1 \right)} + \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) \]
    power algebraApply the power rule to x**2. Simplify the coefficients.✓ Proved
  8. \[ = \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + \left(2 x^{2} - 2 x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) \]
    algebraFactor out log(2*x - 1).✓ Proved
  9. \[ = \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) + \frac{\left(2 x^{2} - 2 x + \frac{1}{2}\right) \frac{d}{d x} \left(2 x - 1\right)}{2 x - 1} \]
    chainApply the chain rule to the logarithm.✓ Proved
  10. \[ = \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) + \frac{2 \left(2 x^{2} - 2 x + \frac{1}{2}\right)}{2 x - 1} \]
    derivativeDifferentiate the inner function 2*x - 1.✓ Proved
  11. \[ = \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) + \frac{4 x^{2} - 4 x + 1}{2 x - 1} \]
    algebraMultiply the terms in the last part.✓ Proved
  12. \[ = - 2 x + \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + 1 + \frac{4 x^{2} - 4 x + 1}{2 x - 1} \]
    derivativeEvaluate the remaining derivatives.✓ Proved
  13. \[ = \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} \]
    algebra simplify simplifyRecognize the perfect square trinomial. Simplify the fraction. Combine all terms.✓ Proved
Answer \( 2 \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} \)
Top: the graph of f = -x² + x + (2x² - 2x + 1/2)·log(2x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2x + (4x - 2)·log(2x - 1) + 1 + 2(2x² - 2x + 1/2)/(2x - 1).

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
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 2*x - 1 = 0
11✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 2*x - 1 = 0
12✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 2*x - 1 = 0
13✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 2*x - 1 = 0
14✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 2*x - 1 = 0
15✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
16✓ 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: fail (style) — Step 7 applies two changes at once (derivative of x**2 and derivative of x) while claiming a single rule application. Each step should modify only one part of the expression.
  • qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and rule applications are accurate.
Every verdict on record (6)
  • qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and rule applications are accurate.
  • gpt-oss:20b: fail (style) 2026-10-18 — Step 7 applies two changes at once (derivative of x**2 and derivative of x) while claiming a single rule application. Each step should modify only one part of the expression.
  • qwen3.6:27b-mlx: pass 2026-10-16
  • gpt-oss:20b: pass 2026-10-16
  • qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.
  • gpt-oss:20b: fail (error) 2026-10-15 — Step 5 applies two rules at once (constant‑multiple and constant) and labels only "constant", violating the one‑rule‑per‑step rule.

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.

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