Derivative of \( \displaystyle \ln{\left(\frac{\sqrt{4 x^{2} - 1}}{\sqrt{1 - 4 x^{2}}} \right)} \)
Problem 2.1455 · hard Beautiful
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\sqrt{4 x^{2} - 1}}{\sqrt{1 - 4 x^{2}}} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\frac{\sqrt{4 x^{2} - 1}}{\sqrt{1 - 4 x^{2}}} \right)} \]algebra logarithmicStart with the derivative of the given function. Use the property log(a/b) = log(a) - log(b) or simplify the quotient first.✓ Proved
- \[ = \frac{d}{d x} \left(- \ln{\left(\sqrt{1 - 4 x^{2}} \right)} + \ln{\left(\sqrt{4 x^{2} - 1} \right)}\right) \]sumSplit the logarithm of the product into the difference of logarithms.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\sqrt{1 - 4 x^{2}} \right)} + \frac{d}{d x} \ln{\left(\sqrt{4 x^{2} - 1} \right)} \]algebra algebraDistribute the derivative operator. Rewrite the square roots as powers.✓ Proved
- \[ = - \frac{d}{d x} \frac{\ln{\left(1 - 4 x^{2} \right)}}{2} + \frac{d}{d x} \frac{\ln{\left(4 x^{2} - 1 \right)}}{2} \]constant-multipleApply the power rule for logarithms.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(1 - 4 x^{2} \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(4 x^{2} - 1 \right)}}{2} \]constant-multipleFactor out the constant 1/2.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(4 x^{2} - 1\right)}{2 \left(4 x^{2} - 1\right)} - \frac{\frac{d}{d x} \left(1 - 4 x^{2}\right)}{2 \left(1 - 4 x^{2}\right)} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{4 x}{4 x^{2} - 1} + \frac{4 x}{1 - 4 x^{2}} \]derivative algebraDifferentiate the inner quadratic functions. Simplify the coefficients.✓ Proved
- \[ = 0 \]algebra simplifyRewrite the second term to have a common denominator. The terms cancel out to zero.✓ Proved
Answer \( 0 \)
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 1 - 4*x**2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 1 - 4*x**2 = 0 |
| 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 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 1 - 4*x**2 = 0 undefined where 4*x**2 - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 4*x**2 = 0 undefined where 4*x**2 - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 4*x**2 = 0 undefined where 4*x**2 - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 4*x**2 = 0 undefined where 4*x**2 - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (style) — Step 2 incorrectly labels an algebraic rewrite as "logarithmic", and step 3 incorrectly labels a logarithmic property as "sum". Both steps apply a single rule, but the rule names are mismatched.qwen3.6:27b-mlx: fail (error) — Step 2 is labeled 'logarithmic' but performs an algebraic rewrite of the quotient into a product using negative exponents, not a logarithmic identity. Step 3 is labeled 'sum' but applies the logarithmic property log(a*b) = log(a) + log(b); the label 'sum' is reserved for linearity of the derivative operator, not algebraic simplification of the integrand.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 2 is labeled 'logarithmic' but performs an algebraic rewrite of the quotient into a product using negative exponents, not a logarithmic identity. Step 3 is labeled 'sum' but applies the logarithmic property log(a*b) = log(a) + log(b); the label 'sum' is reserved for linearity of the derivative operator, not algebraic simplification of the integrand.gpt-oss:20b: fail (style) 2026-10-03 — Step 2 incorrectly labels an algebraic rewrite as "logarithmic", and step 3 incorrectly labels a logarithmic property as "sum". Both steps apply a single rule, but the rule names are mismatched.qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 2 is labeled 'logarithmic' but performs an algebraic rewrite of the quotient into a product with a negative exponent, not the application of a logarithmic derivative rule. Step 3 is labeled 'sum' but applies the logarithmic property log(a*b) = log(a) + log(b) (or log(a/b) = log(a) - log(b)), which is a 'logarithmic' rule, not a linearity of differentiation rule.gpt-oss:20b: fail (error) 2026-10-03 — Step 10 incorrectly changes the sign of the second term; the correct result after applying the chain rule is 4*x/(4*x**2 - 1) - 4*x/(1 - 4*x**2), not a plus. This sign error propagates through the subsequent steps, leading to an incorrect final answer of 0.
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.