Derivative of \( \displaystyle \ln{\left(\sqrt{x \left(2 - x\right)} - \sqrt{x \left(x - 2\right)} \right)} \)
Problem 2.1035 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\sqrt{x \left(2 - x\right)} - \sqrt{x \left(x - 2\right)} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\sqrt{x \left(2 - x\right)} - \sqrt{x \left(x - 2\right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\sqrt{- x^{2} + 2 x} - \sqrt{x^{2} - 2 x} \right)} \]algebra algebraExpand the terms inside the square roots. Rewrite the second square root using a negative sign.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\left(1 - i\right) \sqrt{- x^{2} + 2 x} \right)} \]algebraFactor out the common term.✓ Proved
- \[ = \frac{d}{d x} \left(\ln{\left(\sqrt{- x^{2} + 2 x} \right)} + \ln{\left(\sqrt{2} \right)} - \frac{i \pi}{4}\right) \]rewriteUse the property of logarithms to split the product.✓ Proved
- \[ = \frac{d}{d x} \left(\ln{\left(\sqrt{2} \right)} - \frac{i \pi}{4}\right) + \frac{d}{d x} \ln{\left(\sqrt{- x^{2} + 2 x} \right)} \]sumApply the sum rule for derivatives.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\sqrt{- x^{2} + 2 x} \right)} \]constantThe derivative of a constant is zero.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\left(- x^{2} + 2 x\right)^{0.5} \right)} \]rewriteRewrite the square root as an exponential of a logarithm.≈ Checked numerically
- \[ = \frac{d}{d x} 0.5 \ln{\left(- x^{2} + 2 x \right)} \]chainApply the chain rule to the exponent.≈ Checked numerically
- \[ = 0.5 \frac{d}{d x} \ln{\left(- x^{2} + 2 x \right)} \]constant-multiplePull the constant factor out of the derivative.✓ Proved
- \[ = \frac{0.5 \frac{d}{d x} \left(- x^{2} + 2 x\right)}{- x^{2} + 2 x} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{0.5 \left(2 - 2 x\right)}{- x^{2} + 2 x} \]derivative algebraDifferentiate the inner polynomial. Factor the numerator.✓ Proved
- \[ = \frac{1 - x}{- x^{2} + 2 x} \]simplifySimplify the expression by canceling the constant 2.✓ Proved
Answer \( \frac{x - 1}{x \left(x - 2\right)} \)
Mind the domain. The answer is also defined on (-oo, 0) and (0, 2) and (2, oo), where f(x) is not. Substituting there gives a number that is not a slope of f.
Lines: 13 proved, 2 checked numerically. 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 |
| 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (x - 1)*(-1.0*x*(x - 2) - (x*(2 - x))**1.0)/(x*(2 - x))**2.0; numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments |
| 9 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 1.0*(1 - x)*(-x*(x - 2) - (x*(2 - x))**1.0)/(x*(2 - x))**2.0; numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where -x**2 + 2*x = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where -x**2 + 2*x = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where -x**2 + 2*x = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where -x**2 + 2*x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 0 undefined where x - 2 = 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 8 rewrites log(exp(0.5 log(2*x‑x**2))) as 0.5 log(2*x‑x**2). This simplification is not generally valid (log ∘ exp is not the identity on all complex values), so the subsequent differentiation is based on an incorrect algebraic step.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] Step 4 incorrectly factors the expression inside the logarithm as a product, ignoring that the original expression is a difference of terms (one of which involves complex numbers due to the domain of the square root). Specifically, sqrt(A) - sqrt(-A) cannot be simplified to (1-i)sqrt(A) inside a logarithm without careful handling of branches and the fact that log(a-b) != log(a) + log(b) or similar product rules. The step effectively claims log(sqrt(A) - i*sqrt(A)) = log((1-i)sqrt(A)), which is algebraically true for the argument, but Step 5 then splits log((1-i)sqrt(A)) into log(1-i) + log(sqrt(A)). This split is only valid if the arguments are positive real numbers or under specific branch conventions not generally applicable here, and more importantly, the initial simplification in Step 4 treats the subtraction as a factorization that changes the structure of the function in a way that is not a simple 'algebra' rewrite of the expression form but rather a complex number manipulation that might be valid but is presented as a trivial algebraic step. However, the primary error is in Step 4: `sqrt(x*(2-x)) - sqrt(x*(x-2))` is not equal to `(1-I)*sqrt(2*x - x**2)` for all x in the domain. For real x where 2x-x^2 > 0, `sqrt(x*(x-2))` is imaginary. Let A = 2x-x^2. Then `sqrt(A) - sqrt(-A) = sqrt(A) - i*sqrt(A) = (1-i)sqrt(A)`. This equality holds. But Step 5 uses `log(uv) = log(u) + log(v)`. This is valid for complex logarithms up to branch cuts. The bigger issue is that the problem likely assumes a real domain or standard calculus context where `sqrt(x(x-2))` is undefined for x in (0,2). If the domain is complex, the steps are shaky. If the domain is real, the function is undefined. The prompt says 'Domain (already recorded)... undefined where -x**2 + 2*x = 0'. It doesn't explicitly restrict to real numbers, but standard calculus problems usually imply real domains unless specified. If we assume complex, the step is technically an algebraic rewrite. However, look at Step 9: `Derivative(log(exp(0.5 * log(2*x - x**2))), x)` becomes `Derivative(0.5 * log(2*x - x**2), x)`. This assumes `log(exp(y)) = y`, which is not always true for complex numbers (it equals y + 2piik). This is a significant mathematical defect in a general context. Furthermore, Step 4 is labeled 'algebra' but involves complex arithmetic. The most glaring defect is Step 9's application of the chain rule/log identity without accounting for the branch of the logarithm, or simply assuming `log(exp(z)) = z` which is false for complex z. Given the 'adversarial' instruction, this is an error.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-27 — [domain objection, downgraded to style] Step 4 incorrectly factors the expression inside the logarithm as a product, ignoring that the original expression is a difference of terms (one of which involves complex numbers due to the domain of the square root). Specifically, sqrt(A) - sqrt(-A) cannot be simplified to (1-i)sqrt(A) inside a logarithm without careful handling of branches and the fact that log(a-b) != log(a) + log(b) or similar product rules. The step effectively claims log(sqrt(A) - i*sqrt(A)) = log((1-i)sqrt(A)), which is algebraically true for the argument, but Step 5 then splits log((1-i)sqrt(A)) into log(1-i) + log(sqrt(A)). This split is only valid if the arguments are positive real numbers or under specific branch conventions not generally applicable here, and more importantly, the initial simplification in Step 4 treats the subtraction as a factorization that changes the structure of the function in a way that is not a simple 'algebra' rewrite of the expression form but rather a complex number manipulation that might be valid but is presented as a trivial algebraic step. However, the primary error is in Step 4: `sqrt(x*(2-x)) - sqrt(x*(x-2))` is not equal to `(1-I)*sqrt(2*x - x**2)` for all x in the domain. For real x where 2x-x^2 > 0, `sqrt(x*(x-2))` is imaginary. Let A = 2x-x^2. Then `sqrt(A) - sqrt(-A) = sqrt(A) - i*sqrt(A) = (1-i)sqrt(A)`. This equality holds. But Step 5 uses `log(uv) = log(u) + log(v)`. This is valid for complex logarithms up to branch cuts. The bigger issue is that the problem likely assumes a real domain or standard calculus context where `sqrt(x(x-2))` is undefined for x in (0,2). If the domain is complex, the steps are shaky. If the domain is real, the function is undefined. The prompt says 'Domain (already recorded)... undefined where -x**2 + 2*x = 0'. It doesn't explicitly restrict to real numbers, but standard calculus problems usually imply real domains unless specified. If we assume complex, the step is technically an algebraic rewrite. However, look at Step 9: `Derivative(log(exp(0.5 * log(2*x - x**2))), x)` becomes `Derivative(0.5 * log(2*x - x**2), x)`. This assumes `log(exp(y)) = y`, which is not always true for complex numbers (it equals y + 2piik). This is a significant mathematical defect in a general context. Furthermore, Step 4 is labeled 'algebra' but involves complex arithmetic. The most glaring defect is Step 9's application of the chain rule/log identity without accounting for the branch of the logarithm, or simply assuming `log(exp(z)) = z` which is false for complex z. Given the 'adversarial' instruction, this is an error.gpt-oss:20b: fail (error) 2026-09-27 — Step 8 rewrites log(exp(0.5 log(2*x‑x**2))) as 0.5 log(2*x‑x**2). This simplification is not generally valid (log ∘ exp is not the identity on all complex values), so the subsequent differentiation is based on an incorrect algebraic step.qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 4 incorrectly factors the expression inside the logarithm as a product, ignoring that the original expression is a difference of terms (one of which is imaginary). This algebraic manipulation is invalid for real-valued calculus and leads to an incorrect simplification path.gpt-oss:20b: pass 2026-09-27
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.