Derivative of \( \displaystyle - \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x - 3 \right)} \right)}}{2} \)
Problem 2.1011 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x - 3 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x - 3 \right)} \right)}}{2}\right) \]sumDifferentiate the sum of two terms.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4}\right) + \frac{d}{d x} \frac{\ln{\left(\tan{\left(2 x - 3 \right)} \right)}}{2} \]constantPull out the constant factors.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} + \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x - 3 \right)} \right)}}{2} \]chainApply the chain rule to the logarithms.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(2 x - 3 \right)}}{2 \tan{\left(2 x - 3 \right)}} - \frac{\frac{d}{d x} \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)}{4 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]sumDifferentiate the arguments of the logarithms.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(2 x - 3 \right)}}{2 \tan{\left(2 x - 3 \right)}} - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(2 x - 3 \right)}}{4 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]constantDifferentiate the sum inside the first term.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(2 x - 3 \right)}}{2 \tan{\left(2 x - 3 \right)}} - \frac{\frac{d}{d x} \tan^{2}{\left(2 x - 3 \right)}}{4 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]constantThe derivative of 1 is 0.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(2 x - 3 \right)}}{2 \tan{\left(2 x - 3 \right)}} - \frac{\tan{\left(2 x - 3 \right)} \frac{d}{d x} \tan{\left(2 x - 3 \right)}}{2 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]powerApply the power rule to tan(2*x - 3)**2.✓ Proved
- \[ = \frac{\sec^{2}{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 \tan{\left(2 x - 3 \right)}} - \frac{\tan{\left(2 x - 3 \right)} \sec^{2}{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]chainApply the chain rule to tan(2*x - 3).≈ Checked numerically
- \[ = \frac{\sec^{2}{\left(2 x - 3 \right)}}{\tan{\left(2 x - 3 \right)}} - \frac{\tan{\left(2 x - 3 \right)} \sec^{2}{\left(2 x - 3 \right)}}{\tan^{2}{\left(2 x - 3 \right)} + 1} \]algebra algebra algebraDifferentiate the linear argument 2*x - 3. Simplify the constants and products. Simplify the fractions.✓ Proved
- \[ = - \tan{\left(2 x - 3 \right)} + \frac{\sec^{2}{\left(2 x - 3 \right)}}{\tan{\left(2 x - 3 \right)}} \]algebra simplifySubstitute 1 + tan(2*x - 3)**2 with sec(2*x - 3)**2. Cancel the sec(2*x - 3)**2 term.≈ Checked numerically
- \[ = \frac{\tan^{2}{\left(2 x - 3 \right)} + 1}{\tan{\left(2 x - 3 \right)}} - \tan{\left(2 x - 3 \right)} \]algebraRewrite the second term using the identity for sec^2.≈ Checked numerically
- \[ = \frac{1}{\tan{\left(2 x - 3 \right)}} \]algebra simplifyDistribute the 1/tan(2*x - 3). Combine the tangent terms.✓ Proved
- \[ = \cot{\left(2 x - 3 \right)} \]rewriteRewrite using the cotangent function.✓ Proved
Answer \( \frac{1}{\tan{\left(2 x - 3 \right)}} \)
Lines: 15 proved, 3 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) = 0 undefined where tan(2*x - 3)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3) = 0 undefined where tan(2*x - 3)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3) = 0 undefined where tan(2*x - 3)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3) = 0 undefined where tan(2*x - 3)**2 + 1 = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (tan(2*x - 3)**2 - sec(2*x - 3)**2 + 1)/(tan(2*x - 3)**3 + 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 undefined where tan(2*x - 3)**2 + 1 = 0 sec has poles at odd multiples of pi/2 |
| 9 | ✓ 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 tan(2*x - 3)**2 + 1 = 0 |
| 10 | ✓ 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 tan(2*x - 3)**2 + 1 = 0 |
| 11 | ✓ 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 tan(2*x - 3)**2 + 1 = 0 |
| 12 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (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 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 tan(2*x - 3)**2 + 1 = 0 |
| 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 |
| 14 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-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 sec has poles at odd multiples of pi/2 undefined where tan(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 tan(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 tan(2*x - 3) = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(2*x - 3) = 0 cot has poles at multiples of pi |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 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 (style) — Step 5 incorrectly labels the operation as "constant"; it is a sum rule applied to the derivative of the argument of the logarithm. The label should be "sum". No other multi‑rule steps are present.qwen3.6:27b-mlx: fail (style) — Step 2 is labeled 'constant' but performs the linearity of the derivative (splitting the sum), which should be labeled 'sum'. Step 3 is labeled 'chain' but performs pulling out constant factors, which should be labeled 'constant-multiple'. The labels do not match the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 2 is labeled 'constant' but performs the linearity of the derivative (splitting the sum), which should be labeled 'sum'. Step 3 is labeled 'chain' but performs pulling out constant factors, which should be labeled 'constant-multiple'. The labels do not match the operations performed.gpt-oss:20b: fail (style) 2026-09-27 — Step 5 incorrectly labels the operation as "constant"; it is a sum rule applied to the derivative of the argument of the logarithm. The label should be "sum". No other multi‑rule steps are present.qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 2 applies the constant multiple rule to both terms simultaneously, violating the one-change-per-step constraint. Step 4 applies the chain rule to both terms simultaneously, also violating the constraint.gpt-oss:20b: fail (error) 2026-09-27 — Step 4 incorrectly labels the chain rule as "sum". Step 5 also mislabels the application of the sum rule as "constant". These mislabelings violate the contract that each step must name the exact rule applied.
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.