Derivative of \( \displaystyle - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} + \frac{\ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \)
Problem 2.2013 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} + \frac{\ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} + \frac{\ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6}\right) + \frac{d}{d x} \frac{\ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} + \frac{\frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \]constant-multipleFactor out the constants.✓ Proved
- \[ = - \frac{\cos^{2}{\left(3 x - 3 \right)} \frac{d}{d x} \frac{1}{\cos^{2}{\left(3 x - 3 \right)}}}{6} + \frac{\frac{d}{d x} \tan{\left(3 x - 3 \right)}}{3 \tan{\left(3 x - 3 \right)}} \]logarithmicApply the derivative rule for natural logs.✓ Proved
- \[ = \frac{\sec^{2}{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right)}{3 \tan{\left(3 x - 3 \right)}} + \frac{\frac{d}{d x} \cos{\left(3 x - 3 \right)}}{3 \cos{\left(3 x - 3 \right)}} \]chainApply the chain rule to the inner functions.✓ Proved
- \[ = - \frac{\sin{\left(3 x - 3 \right)}}{\cos{\left(3 x - 3 \right)}} + \frac{\sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]chain algebra algebra algebraDifferentiate the innermost linear function. Multiply the constants in the numerators. Simplify the fraction involving powers of cosine. Rewrite the negative exponent as a fraction.✓ Proved
- \[ = - \tan{\left(3 x - 3 \right)} + \frac{\sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]trigUse the definition of tangent.✓ Proved
- \[ = - \tan{\left(3 x - 3 \right)} + \frac{1}{\sin{\left(3 x - 3 \right)} \cos{\left(3 x - 3 \right)}} \]trig algebra algebraUse the definition of secant. Simplify the complex fraction. Introduce a factor of 2 to prepare for the double angle identity.✓ Proved
- \[ = - \tan{\left(3 x - 3 \right)} + \frac{2}{\sin{\left(6 x - 6 \right)}} \]algebraUse the double angle identity for sine.✓ Proved
- \[ = - \tan{\left(3 x - 3 \right)} + 2 \csc{\left(6 x - 6 \right)} \]trig algebraUse the definition of cosecant. Distribute the 3 inside the argument.✓ Proved
- \[ = - \tan{\left(3 x - 3 \right)} + \frac{1}{\sin{\left(3 x - 3 \right)} \cos{\left(3 x - 3 \right)}} \]algebra algebraExpand the argument back to its basic form to simplify. Cancel the 2.✓ Proved
- \[ = - \tan{\left(3 x - 3 \right)} + \frac{\sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]algebraRevert to secant form for final simplification.✓ Proved
- \[ = - \tan{\left(3 x - 3 \right)} + \frac{1}{\sin{\left(3 x - 3 \right)} \cos{\left(3 x - 3 \right)}} \]algebraWait, let's use the simplest form directly.✓ Proved
- \[ = - \tan{\left(3 x - 3 \right)} + 2 \csc{\left(6 x - 6 \right)} \]simplifyThe simplest form is the result of step 15.✓ Proved
Answer \( \frac{1}{\tan{\left(3 x - 3 \right)}} \)
Lines: 21 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 undefined where tan(3*x - 3) = 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(3*x - 3) = 0 undefined where cos(3*x - 3) = 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(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 |
| 7 | ✓ 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(3*x - 3) = 0 undefined where cos(3*x - 3) = 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(3*x - 3) = 0 undefined where cos(3*x - 3) = 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(3*x - 3) = 0 undefined where cos(3*x - 3) = 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(3*x - 3) = 0 undefined where cos(3*x - 3) = 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(3*x - 3) = 0 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 tan has poles at odd multiples of pi/2 undefined where sin(3*x - 3) = 0 undefined where cos(3*x - 3) = 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(3*x - 3) = 0 undefined where cos(3*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(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 undefined where sin(6*x - 6) = 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(6*x - 6) = 0 csc has poles at multiples of pi |
| 16 | ✓ 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 |
| 17 | ✓ 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(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 sec has poles at odd multiples of pi/2 undefined where tan(3*x - 3) = 0 |
| 20 | ✓ 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(3*x - 3) = 0 undefined where sin(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 |
| 21 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(3*x - 3) = 0 undefined where cos(3*x - 3) = 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(3*x - 3) - 1/tan(3*x - 3) + 2/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 (style) — Step 17 applies two transformations at once (rewriting csc to 1/sin and expanding the argument), violating the one‑rule‑per‑step rule. Additionally, step 21 jumps back to an earlier form without a valid justification, making the sequence inconsistent.qwen3.6:27b-mlx: fail (error) — The final answer provided in the 'Stated answer' field (1/tan(3*x - 3)) does not match the result of the derivation steps (-tan(3*x - 3) + 2*csc(6*x - 6)). The derivation is mathematically correct for the given function, but the stated answer is incorrect, creating a contradiction.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-10 — Step 17 applies two transformations at once (rewriting csc to 1/sin and expanding the argument), violating the one‑rule‑per‑step rule. Additionally, step 21 jumps back to an earlier form without a valid justification, making the sequence inconsistent.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The final answer provided in the 'Stated answer' field (1/tan(3*x - 3)) does not match the result of the derivation steps (-tan(3*x - 3) + 2*csc(6*x - 6)). The derivation is mathematically correct for the given function, but the stated answer is incorrect, creating a contradiction.gpt-oss:20b: fail (error) 2026-10-10 — The final simplification to 2*csc(6*x-6) is incorrect; it does not equal the earlier expression -tan(3*x-3)+sec(3*x-3)**2/tan(3*x-3) which simplifies to 1/tan(3*x-3).qwen3.6:27b-mlx: fail (error) 2026-10-10 — The final answer provided in the task description (1/tan(3*x - 3)) is mathematically incorrect; the correct derivative simplifies to 2*csc(6*x - 6). The solution steps correctly derive the latter but fail to match the stated answer, and the final steps (17-21) are incoherent, reverting to previous forms and contradicting themselves.
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 2026-10-10 with SymPy 1.14.0.