Derivative of \( \displaystyle - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{2} + \ln{\left(\tan{\left(3 x - 3 \right)} \right)} \)
Problem 2.1409 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{2} + \ln{\left(\tan{\left(3 x - 3 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{2} + \ln{\left(\tan{\left(3 x - 3 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{2}\right) + \frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{2} + \frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)} \]constant-multipleFactor out the constant -1/2.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(- 2 \ln{\left(\cos{\left(3 x - 3 \right)} \right)}\right)}{2} + \frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)} \]logarithmicUse the property log(a^b) = b*log(a).✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)} + \frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)} \]algebraSimplify the constant coefficients: (-1/2) * (-2) = 1.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)} + \frac{\frac{d}{d x} \tan{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]chainApply the chain rule to the second term.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)} + \frac{3 \sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]trig algebraDifferentiate the tangent function. Rewrite the fraction.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)} + \frac{3}{\sin{\left(3 x - 3 \right)} \cos{\left(3 x - 3 \right)}} \]rewrite algebraExpress secant in terms of cosine and tangent in terms of sine and cosine. Simplify the denominator.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)} + \frac{3.0}{\sin{\left(3 x - 3 \right)} \cos{\left(3 x - 3 \right)}} \]algebraPrepare for double angle identity.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)} + \frac{6.0}{\sin{\left(6 x - 6 \right)}} \]algebraUse the double angle identity 2*sin(u)*cos(u) = sin(2u).✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)} + \frac{6}{\sin{\left(6 x - 6 \right)}} \]algebraSimplify the fraction.✓ Proved
- \[ = 6 \csc{\left(6 x - 6 \right)} + \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)} \]rewriteRewrite 1/sin(u) as csc(u).✓ Proved
- \[ = - \frac{3 \sin{\left(3 x - 3 \right)}}{\cos{\left(3 x - 3 \right)}} + 6 \csc{\left(6 x - 6 \right)} \]chainDifferentiate the first term using the chain rule.✓ Proved
- \[ = - 3 \tan{\left(3 x - 3 \right)} + 6 \csc{\left(6 x - 6 \right)} \]algebraSimplify the first term using tan(u) = sin(u)/cos(u).✓ Proved
Answer \( \frac{3}{\tan{\left(3 x - 3 \right)}} \)
Lines: 16 proved, 1 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 undefined where cos(3*x - 3) = 0 |
| 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 cos(3*x - 3) = 0 |
| 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 cos(3*x - 3) = 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 |
| 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(3*x - 3) = 0 |
| 7 | ✓ 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(3*x - 3) = 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(3*x - 3) = 0 |
| 9 | ✓ 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(3*x - 3) = 0 undefined where sin(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 undefined where sin(6*x - 6) = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(6*x - 6) = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(6*x - 6) = 0 csc has poles at multiples of pi |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments csc has poles at multiples of pi undefined where cos(3*x - 3) = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi undefined where cos(3*x - 3) = 0 tan has poles at odd multiples of pi/2 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -3*tan(3*x - 3) - 3/tan(3*x - 3) + 6/sin(6*x - 6); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(3*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) — The final derivative is incorrect. The correct derivative of f(x) is 3/tan(3*x-3), but the solution produces -3*tan(3*x-3)+6*csc(6*x-6). The steps also leave the derivative of log(cos(3*x-3)) unevaluated, leading to an incomplete result.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — The final derivative is incorrect. The correct derivative of f(x) is 3/tan(3*x-3), but the solution produces -3*tan(3*x-3)+6*csc(6*x-6). The steps also leave the derivative of log(cos(3*x-3)) unevaluated, leading to an incomplete result.qwen3.6:27b-mlx: fail (error) 2026-10-03 — The final answer provided in the task description (3/tan(3*x - 3)) is mathematically incorrect; the correct derivative is 6*csc(6*x - 6) - 3*tan(3*x - 3). The step-by-step derivation correctly arrives at the latter, but the solution fails to match the stated answer, indicating a defect in the problem statement or the final claim.gpt-oss:20b: inconclusive 2026-10-03 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly simplifies the derivative of \u201clog(tan(3x-3))\u201d. It replaces \u2013sec^2(3x-3)/tan(3x-3) with 6\,csc(6x-6), which is alge
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.