Derivative of \( \displaystyle - \frac{\ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} + \frac{\ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4} \)
Problem 2.1426 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} + \frac{\ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} + \frac{\ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8}\right) + \frac{d}{d x} \frac{\ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4} \]sumSplit the derivative into two parts.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} + \frac{\frac{d}{d x} \ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4} \]constantPull out the constant coefficients.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(4 x - 1 \right)}}{4 \tan{\left(4 x - 1 \right)}} - \frac{\frac{d}{d x} \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]chainApply the chain rule to both terms.✓ Proved
- \[ = \frac{\sec^{2}{\left(4 x - 1 \right)} \frac{d}{d x} \left(4 x - 1\right)}{4 \tan{\left(4 x - 1 \right)}} - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(4 x - 1 \right)}}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]sumApply the sum rule inside the first derivative.≈ Checked numerically
- \[ = \frac{\sec^{2}{\left(4 x - 1 \right)}}{\tan{\left(4 x - 1 \right)}} - \frac{\tan{\left(4 x - 1 \right)} \frac{d}{d x} \tan{\left(4 x - 1 \right)}}{4 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]powerDifferentiate the power term using the power rule.✓ Proved
- \[ = \frac{\sec^{2}{\left(4 x - 1 \right)}}{\tan{\left(4 x - 1 \right)}} - \frac{\tan{\left(4 x - 1 \right)} \sec^{2}{\left(4 x - 1 \right)}}{\tan^{2}{\left(4 x - 1 \right)} + 1} \]chain algebra algebraApply the chain rule to the tangent term. Simplify the products and constants. Simplify the fractions.≈ Checked numerically
- \[ = - \tan{\left(4 x - 1 \right)} + \frac{\sec^{2}{\left(4 x - 1 \right)}}{\tan{\left(4 x - 1 \right)}} \]algebra algebraUse the identity tan(u)**2 + 1 = sec(u)**2. Cancel the sec(u)**2 term in the first fraction.≈ Checked numerically
- \[ = - \tan{\left(4 x - 1 \right)} + \frac{1}{\sin{\left(4 x - 1 \right)} \cos{\left(4 x - 1 \right)}} \]rewrite algebra algebraRewrite sec(u)**2 as 1/cos(u)**2. Simplify the complex fraction. Introduce a 2 in the numerator and denominator.✓ Proved
- \[ = - \tan{\left(4 x - 1 \right)} + \frac{2}{\sin{\left(8 x - 2 \right)}} \]algebraUse the double angle identity sin(2u) = 2sin(u)cos(u).✓ Proved
- \[ = - \tan{\left(4 x - 1 \right)} + 2 \csc{\left(8 x - 2 \right)} \]simplifyUse the identity 1/sin(u) = csc(u) and simplify the argument.✓ Proved
Answer \( \frac{1}{\tan{\left(4 x - 1 \right)}} \)
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: 13 proved, 4 checked numerically. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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(4*x - 1)**2 + 1 = 0 undefined where tan(4*x - 1) = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (tan(4*x - 1)**2 - sec(4*x - 1)**2 + 1)/tan(4*x - 1); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(4*x - 1)**2 + 1 = 0 undefined where tan(4*x - 1) = 0 sec has poles at odd multiples of pi/2 |
| 6 | ✓ 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(4*x - 1)**2 + 1 = 0 undefined where tan(4*x - 1) = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-tan(4*x - 1)**2 + sec(4*x - 1)**2 - 1)*tan(4*x - 1)/(tan(4*x - 1)**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(4*x - 1)**2 + 1 = 0 undefined where tan(4*x - 1) = 0 |
| 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(4*x - 1)**2 + 1 = 0 undefined where tan(4*x - 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(4*x - 1)**2 + 1 = 0 undefined where tan(4*x - 1) = 0 |
| 10 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (tan(4*x - 1)**2 - sec(4*x - 1)**2 + 1)*tan(4*x - 1)/(tan(4*x - 1)**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(4*x - 1)**2 + 1 = 0 undefined where tan(4*x - 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(4*x - 1) = 0 |
| 12 | ✓ 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(4*x - 1) = 0 undefined where sin(4*x - 1) = 0 undefined where cos(4*x - 1) = 0 |
| 13 | ✓ 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 - 1) = 0 undefined where cos(4*x - 1) = 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(4*x - 1) = 0 undefined where cos(4*x - 1) = 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(4*x - 1) = 0 undefined where cos(4*x - 1) = 0 undefined where sin(8*x - 2) = 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(8*x - 2) = 0 csc has poles at multiples of pi |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -tan(4*x - 1) + 2*csc(8*x - 2) - 1/tan(4*x - 1); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(4*x - 1) = 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) — The final result "1/tan(4*x - 1)" is incorrect. The correct derivative simplifies to "-tan(4*x - 1) + 2*csc(8*x - 2)" (or an equivalent form).qwen3.6:27b-mlx: fail (error) — Step 4 applies the chain rule to both terms simultaneously, violating the 'one rule per step' constraint. Additionally, Step 6 labels the differentiation of tan(u)^2 as 'power', but this step also requires the chain rule for the inner function tan(u), which is not accounted for in the label or the step's logic (the derivative of tan is not applied until Step 7, making Step 6 incomplete or mislabeled).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 4 applies the chain rule to both terms simultaneously, violating the 'one rule per step' constraint. Additionally, Step 6 labels the differentiation of tan(u)^2 as 'power', but this step also requires the chain rule for the inner function tan(u), which is not accounted for in the label or the step's logic (the derivative of tan is not applied until Step 7, making Step 6 incomplete or mislabeled).gpt-oss:20b: fail (error) 2026-10-03 — The final result "1/tan(4*x - 1)" is incorrect. The correct derivative simplifies to "-tan(4*x - 1) + 2*csc(8*x - 2)" (or an equivalent form).qwen3.6:27b-mlx: fail (error) 2026-10-03 — The final answer is mathematically incorrect; the derivative simplifies to 0, not 1/tan(4*x - 1). The error occurs in Step 11, where the term sec(4*x - 1)**2 / tan(4*x - 1) is incorrectly simplified to 1/tan(4*x - 1) instead of tan(4*x - 1), leading to -tan + 1/tan instead of -tan + tan = 0.gpt-oss:20b: inconclusive 2026-10-03 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The simplification from step 10 onward is incorrect. From \n\(-\tan(4x-1)+\sec^2(4x-1)/\tan(4x-1)\) one should obtain \((\sec^2-\tan^2)/\tan=1/\tan\), no
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.