Derivative of \( \displaystyle \ln{\left(2 x + 2 \sqrt{x \left(x - 1\right)} - 1 \right)} \)
Problem 2.1032 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(2 x + 2 \sqrt{x \left(x - 1\right)} - 1 \right)} \).
- \[ \frac{d}{d x} \ln{\left(2 x + 2 \sqrt{x \left(x - 1\right)} - 1 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(2 x + 2 \sqrt{x \left(x - 1\right)} - 1\right)}{2 x + 2 \sqrt{x \left(x - 1\right)} - 1} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{- \frac{d}{d x} 1 + \frac{d}{d x} 2 x + \frac{d}{d x} 2 \sqrt{x \left(x - 1\right)}}{2 x + 2 \sqrt{x \left(x - 1\right)} - 1} \]sumDifferentiate the sum of terms.✓ Proved
- \[ = \frac{\frac{d}{d x} 2 \sqrt{x \left(x - 1\right)} + 2}{2 x + 2 \sqrt{x \left(x - 1\right)} - 1} \]constantDifferentiate the constant term.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \sqrt{x \left(x - 1\right)} + 2}{2 x + 2 \sqrt{x \left(x - 1\right)} - 1} \]constant-multipleFactor out the constant 2.✓ Proved
- \[ = \frac{2 + \frac{\frac{d}{d x} x \left(x - 1\right)}{\sqrt{x \left(x - 1\right)}}}{2 x + 2 \sqrt{x \left(x - 1\right)} - 1} \]chainApply the chain rule to the square root.✓ Proved
- \[ = \frac{2 + \frac{x \frac{d}{d x} \left(x - 1\right) + \left(x - 1\right) \frac{d}{d x} x}{\sqrt{x \left(x - 1\right)}}}{2 x + 2 \sqrt{x \left(x - 1\right)} - 1} \]productApply the product rule to the radicand.✓ Proved
- \[ = \frac{2 + \frac{2 x - 1}{\sqrt{x \left(x - 1\right)}}}{2 x + 2 \sqrt{x \left(x - 1\right)} - 1} \]derivative algebra algebraDifferentiate the terms inside the product rule. Simplify the expression inside the parenthesis. Simplify the fraction by canceling the 2.✓ Proved
- \[ = \frac{1}{\sqrt{x \left(x - 1\right)}} \]algebra algebraFind a common denominator for the numerator. Multiply the denominator terms.✓ Proved
- \[ = \frac{2 x + 2 \sqrt{x \left(x - 1\right)} - 1}{2 x \sqrt{x \left(x - 1\right)} + 2 x \left(x - 1\right) - \sqrt{x \left(x - 1\right)}} \]algebraDistribute the square root term in the denominator.✓ Proved
- \[ = \frac{1}{\sqrt{x \left(x - 1\right)}} \]simplifySimplify the expression to its final form.✓ Proved
Answer \( \frac{1}{\sqrt{x \left(x - 1\right)}} \)
Mind the domain. The answer is also defined on (-oo, 0), where f(x) is not. Substituting there gives a number that is not a slope of f.
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.
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 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 undefined where x*(x - 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 undefined where x*(x - 1) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 undefined where x*(x - 1) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 undefined where x*(x - 1) = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 undefined where x*(x - 1) = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2*sqrt(x*(x - 1)) - 1 = 0 undefined where x*(x - 1) = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x*(x - 1) = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x*(x - 1) = 0 undefined where 2*x*sqrt(x*(x - 1)) + 2*x*(x - 1) - sqrt(x*(x - 1)) = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x*sqrt(x*(x - 1)) + 2*x*(x - 1) - sqrt(x*(x - 1)) = 0 undefined where x*(x - 1) = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x*(x - 1) = 0 |
| 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 14 claims the fraction simplifies to 1/√(x(x‑1)), but the algebraic simplification is incorrect. The expression (2√(x(x‑1))+2x‑1)/(2x√(x(x‑1))+2x(x‑1)‑√(x(x‑1))) does not reduce to 1/√(x(x‑1)).qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. The final simplification is algebraically valid and correctly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. The final simplification is algebraically valid and correctly labeled.gpt-oss:20b: fail (error) 2026-09-27 — Step 14 claims the fraction simplifies to 1/√(x(x‑1)), but the algebraic simplification is incorrect. The expression (2√(x(x‑1))+2x‑1)/(2x√(x(x‑1))+2x(x‑1)‑√(x(x‑1))) does not reduce to 1/√(x(x‑1)).qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. The algebraic simplification in the final steps is valid and leads to the correct result.gpt-oss:20b: fail (error) 2026-09-27 — Step 14 incorrectly simplifies the fraction; the expression (2√(x(x‑1))+2x‑1)/(2x√(x(x‑1))+2x(x‑1)‑√(x(x‑1))) does not reduce to 1/√(x(x‑1)).
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 2026-09-27 with SymPy 1.14.0.