Derivative of \( \displaystyle - \frac{\ln{\left(1 - x^{2} \right)}}{2} + \frac{\ln{\left(x^{2} + 1 \right)}}{2} \)
Problem 2.1365 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(1 - x^{2} \right)}}{2} + \frac{\ln{\left(x^{2} + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(1 - x^{2} \right)}}{2} + \frac{\ln{\left(x^{2} + 1 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{d}{d x} \frac{\ln{\left(x^{2} + 1 \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]algebra constant-multipleSimplify the expression inside the first logarithm. Factor out the constant 1/2 from the second term.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} - \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \left(1 - x^{2}\right)} \]chainApply the chain rule to both logarithmic terms.✓ Proved
- \[ = \frac{x}{x^{2} + 1} + \frac{x}{1 - x^{2}} \]derivative constant-multiple algebraDifferentiate the inner functions. Simplify the constants in the fractions. Simplify the signs.✓ Proved
- \[ = \frac{x \left(1 - x^{2}\right) + x \left(x^{2} + 1\right)}{\left(1 - x^{2}\right) \left(x^{2} + 1\right)} \]algebraFind a common denominator.✓ Proved
- \[ = \frac{2 x}{1 - x^{4}} \]algebra simplifyExpand the numerator and denominator. Combine like terms in the numerator.✓ Proved
Answer \( - \frac{2 x}{x^{4} - 1} \)
Mind the domain. The answer is also defined on (-oo, -1) and (1, oo), 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 |
| 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 undefined where 1 - x**2 = 0 undefined where x**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - x**2 = 0 undefined where x**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - x**2 = 0 undefined where x**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - x**2 = 0 undefined where x**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - x**2 = 0 undefined where x**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - x**2 = 0 undefined where x**2 + 1 = 0 undefined where 1 - x**4 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - x**4 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x**4 - 1 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Step 2 applies the sum rule but incorrectly distributes the negative sign and constant factor, resulting in an algebraically incorrect expression (the first term should be -1/2 * Derivative(log(1-x^2), x), but the step writes -Derivative(log(1-x^2), x)/2 which is correct, however the second term changes from log(x^2+1)/2 to Derivative(log(x^2+1)/2, x) while the first term's structure is altered inconsistently with the 'sum' rule label which implies linearity. More critically, Step 5 applies the chain rule to BOTH terms simultaneously, violating the 'one rule at a time' constraint. Step 2 also fails to properly isolate the constant multiple for the first term, mixing sum and constant-multiple rules.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 2 applies the sum rule but incorrectly distributes the negative sign and constant factor, resulting in an algebraically incorrect expression (the first term should be -1/2 * Derivative(log(1-x^2), x), but the step writes -Derivative(log(1-x^2), x)/2 which is correct, however the second term changes from log(x^2+1)/2 to Derivative(log(x^2+1)/2, x) while the first term's structure is altered inconsistently with the 'sum' rule label which implies linearity. More critically, Step 5 applies the chain rule to BOTH terms simultaneously, violating the 'one rule at a time' constraint. Step 2 also fails to properly isolate the constant multiple for the first term, mixing sum and constant-multiple rules.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 5 applies the chain rule to both terms simultaneously, violating the one-rule-per-step constraint. Step 10 incorrectly simplifies the denominator (1 - x^2)(x^2 + 1) to 1 - x^4, whereas it should be 1 + x^2 - x^2 - x^4 = 1 - x^4 is actually correct algebraically, but wait: (1-x^2)(1+x^2) = 1 - x^4. Let me re-read step 10. Numerator: x(x^2+1) + x(1-x^2) = x^3 + x + x - x^3 = 2x. Denominator: (1-x^2)(x^2+1) = 1 + x^2 - x^2 - x^4 = 1 - x^4. So step 10 is algebraically correct. However, the final answer in the prompt is -2x/(x^4-1). Step 11 gives 2x/(1-x^4). Note that 2x/(1-x^4) = -2x/(x^4-1). So the math is correct. The defect is Step 5 applying the rule to two terms at once.gpt-oss:20b: fail (error) 2026-09-30 — Step 11 incorrectly simplifies the fraction: (2*x)/(1 - x**4) is equal to -2*x/(x**4 - 1), so the final result should have a negative sign. The solution omits this sign change, leading to an incorrect derivative.
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 2026-10-03 with SymPy 1.14.0.