Derivative of \( \displaystyle \frac{\ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} \)
Problem 2.192 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} \]constant-multiplePull out the constant factor 1/4.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)}{4 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]chainApply the chain rule for the natural logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(2 x - 1 \right)}}{4 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]sumApply the sum rule to the inner expression.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan^{2}{\left(2 x - 1 \right)}}{4 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]constantThe derivative of the constant 1 is 0.✓ Proved
- \[ = \frac{\tan{\left(2 x - 1 \right)} \frac{d}{d x} \tan{\left(2 x - 1 \right)}}{2 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]powerApply the power rule to the squared tangent term.✓ Proved
- \[ = \frac{\tan{\left(2 x - 1 \right)} \sec^{2}{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{2 \left(\tan^{2}{\left(2 x - 1 \right)} + 1\right)} \]chainApply the chain rule to the tangent function.≈ Checked numerically
- \[ = \frac{\tan{\left(2 x - 1 \right)} \sec^{2}{\left(2 x - 1 \right)}}{\tan^{2}{\left(2 x - 1 \right)} + 1} \]derivative algebra simplifyThe derivative of 2*x - 1 is 2. Multiply the constants together. Simplify the expression using the identity 1 + tan(u)^2 = sec(u)^2.✓ Proved
Answer \( \tan{\left(2 x - 1 \right)} \)
Mind the domain. The answer is also defined at points where f(x) is not. Substituting there gives a number that is not a slope of f.
Lines: 9 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 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 undefined where tan(2*x - 1)**2 + 1 = 0 |
| 4 | ✓ 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 - 1)**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 - 1)**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 - 1)**2 + 1 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (tan(2*x - 1)**2 - sec(2*x - 1)**2 + 1)*tan(2*x - 1)/(tan(2*x - 1)**2 + 1); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(2*x - 1)**2 + 1 = 0 sec has poles at odd multiples of pi/2 |
| 8 | ✓ 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 - 1)**2 + 1 = 0 |
| 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 - 1)**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 - 1)**2 + 1 = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left (-tan(2*x - 1)**2 + sec(2*x - 1)**2 - 1)*tan(2*x - 1)/(tan(2*x - 1)**2 + 1); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passdeepseek-r1:70b: fail (style) — Step 1 is missing a rule label.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final simplification using the trigonometric identity is valid and correctly labeled.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final simplification using the trigonometric identity is valid and correctly labeled.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final simplification using the trigonometric identity is valid and correctly labeled.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the single-change constraint and using valid labels from the fixed vocabulary.deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 is missing a rule label.gpt-oss:20b: inconclusive 2026-09-19 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 6 applies both the power rule and the chain rule at once, producing an incorrect intermediate expression. The derivative of \(\tan(2x-1)^2\) should bqwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 lacks a rule label, which is required by the contract.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final simplification using the trigonometric identity is valid and correctly labeled.deepseek-r1:70b: fail (style) 2026-09-18 — Step 1 is missing a required rule label.gpt-oss:20b: pass 2026-09-18gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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 2026-09-26 with SymPy 1.14.0.