Derivative of \( \displaystyle - \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{2} + \ln{\left(\tan{\left(2 x - 3 \right)} \right)} \)
Problem 2.1196 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{2} + \ln{\left(\tan{\left(2 x - 3 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{2} + \ln{\left(\tan{\left(2 x - 3 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{2}\right) + \frac{d}{d x} \ln{\left(\tan{\left(2 x - 3 \right)} \right)} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{2} + \frac{d}{d x} \ln{\left(\tan{\left(2 x - 3 \right)} \right)} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(2 x - 3 \right)} \right)} - \frac{\frac{d}{d x} \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)}{2 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(2 x - 3 \right)} \right)} - \frac{\frac{d}{d x} \tan^{2}{\left(2 x - 3 \right)}}{2 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]derivativeDifferentiate the constant term inside.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(2 x - 3 \right)} \right)} - \frac{\tan{\left(2 x - 3 \right)} \frac{d}{d x} \tan{\left(2 x - 3 \right)}}{\tan^{2}{\left(2 x - 3 \right)} + 1} \]powerApply the power rule.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(2 x - 3 \right)} \right)} - \frac{2 \tan{\left(2 x - 3 \right)} \sec^{2}{\left(2 x - 3 \right)}}{\tan^{2}{\left(2 x - 3 \right)} + 1} \]chain algebraDifferentiate the tangent function. Simplify the coefficients.≈ Checked numerically
- \[ = - 2 \tan{\left(2 x - 3 \right)} + \frac{d}{d x} \ln{\left(\tan{\left(2 x - 3 \right)} \right)} \]algebra simplifyUse the identity 1 + tan(u)^2 = sec(u)^2. Cancel the secant squared terms.≈ Checked numerically
- \[ = - 2 \tan{\left(2 x - 3 \right)} + \frac{\frac{d}{d x} \tan{\left(2 x - 3 \right)}}{\tan{\left(2 x - 3 \right)}} \]chainApply the chain rule to the second logarithm.✓ Proved
- \[ = - 2 \tan{\left(2 x - 3 \right)} + \frac{2 \sec^{2}{\left(2 x - 3 \right)}}{\tan{\left(2 x - 3 \right)}} \]derivativeDifferentiate the tangent function.≈ Checked numerically
- \[ = - 2 \tan{\left(2 x - 3 \right)} + \frac{2}{\sin{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)}} \]algebra simplify algebraRewrite secant in terms of cosine. Simplify the fraction by canceling one cosine term. Introduce a factor of 2 to use the double angle identity.✓ Proved
- \[ = - 2 \tan{\left(2 x - 3 \right)} + \frac{4}{\sin{\left(4 x - 6 \right)}} \]algebraApply the double angle identity for sine.✓ Proved
- \[ = - 2 \tan{\left(2 x - 3 \right)} + 4 \csc{\left(4 x - 6 \right)} \]rewriteConvert sine to cosecant.✓ Proved
- \[ = - 2 \tan{\left(2 x - 3 \right)} + \frac{4}{\sin{\left(4 x - 6 \right)}} \]simplifyFinal simplification.✓ Proved
Answer \( \frac{2}{\tan{\left(2 x - 3 \right)}} \)
Lines: 15 proved, 4 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 tan has poles at odd multiples of pi/2 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3)**2 + 1 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*(-tan(2*x - 3)**2 + sec(2*x - 3)**2 - 1)*tan(2*x - 3)/(tan(2*x - 3)**2 + 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3)**2 + 1 = 0 sec has poles at odd multiples of pi/2 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(2*x - 3)**2 + 1 = 0 |
| 9 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*(tan(2*x - 3)**2 - sec(2*x - 3)**2 + 1)*tan(2*x - 3)/(tan(2*x - 3)**2 + 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(2*x - 3)**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3) = 0 |
| 12 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*(tan(2*x - 3)**2 - sec(2*x - 3)**2 + 1)/tan(2*x - 3); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3) = 0 sec has poles at odd multiples of pi/2 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(2*x - 3) = 0 undefined where sin(2*x - 3) = 0 undefined where cos(2*x - 3) = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(2*x - 3) = 0 undefined where cos(2*x - 3) = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(2*x - 3) = 0 undefined where cos(2*x - 3) = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(2*x - 3) = 0 undefined where cos(2*x - 3) = 0 undefined where sin(4*x - 6) = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(4*x - 6) = 0 csc has poles at multiples of pi |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 csc has poles at multiples of pi undefined where sin(4*x - 6) = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -2*tan(2*x - 3) - 2/tan(2*x - 3) + 4/sin(4*x - 6); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3) = 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 incorrectly simplifies the coefficient: -1/2 * (2 * tan(2*x - 3) * sec(2*x - 3)**2 / (tan(2*x - 3)**2 + 1)) should equal -tan(2*x - 3) * sec(2*x - 3)**2 / (tan(2*x - 3)**2 + 1), not -2 * tan(2*x - 3) * sec(2*x - 3)**2 / (tan(2*x - 3)**2 + 1).qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: fail (error) 2026-09-29 — Step 8 incorrectly simplifies the coefficient: -1/2 * (2 * tan(2*x - 3) * sec(2*x - 3)**2 / (tan(2*x - 3)**2 + 1)) should equal -tan(2*x - 3) * sec(2*x - 3)**2 / (tan(2*x - 3)**2 + 1), not -2 * tan(2*x - 3) * sec(2*x - 3)**2 / (tan(2*x - 3)**2 + 1).qwen3.6:27b-mlx: fail (error) 2026-09-29 — The final answer is mathematically incorrect. The derivative of the first term is -2*tan(2x-3), and the derivative of the second term is 2*sec(2x-3)^2/tan(2x-3). These do not simplify to 2/tan(2x-3). Specifically, 2*sec^2(u)/tan(u) = 2/(sin(u)cos(u)) = 4/sin(2u), which is not equal to 2/tan(u) in general. The solution fails to combine the terms correctly or simplify to the stated answer.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.