Derivative of \( \displaystyle - \frac{5 \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} + \frac{5 \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \)
Problem 2.972 · hard
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} + \frac{5 \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} + \frac{5 \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3}\right) \]sumDifferentiate the sum of two terms.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6}\right) + \frac{d}{d x} \frac{5 \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \]sumSplit the derivative into two parts.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} + \frac{5 \frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \]constant-multiplePull out the constant factors.✓ Proved
- \[ = - \frac{5 \cos^{2}{\left(3 x - 3 \right)} \frac{d}{d x} \frac{1}{\cos^{2}{\left(3 x - 3 \right)}}}{6} + \frac{5 \frac{d}{d x} \tan{\left(3 x - 3 \right)}}{3 \tan{\left(3 x - 3 \right)}} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{5 \sec^{2}{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right)}{3 \tan{\left(3 x - 3 \right)}} + \frac{5 \frac{d}{d x} \cos{\left(3 x - 3 \right)}}{3 \cos{\left(3 x - 3 \right)}} \]chainApply the chain rule to the power and trigonometric terms.✓ Proved
- \[ = - \frac{5 \sin{\left(3 x - 3 \right)}}{\cos{\left(3 x - 3 \right)}} + \frac{5 \sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]chain algebra algebra algebraDifferentiate the innermost functions. Simplify the products in the numerator. Simplify the constant coefficients. Combine the powers of cosine.✓ Proved
- \[ = - \frac{5 \sin{\left(3 x - 3 \right)}}{\cos{\left(3 x - 3 \right)}} + 5 \cot{\left(3 x - 3 \right)} \sec^{2}{\left(3 x - 3 \right)} \]algebraRewrite the expression using tangent and cotangent.✓ Proved
- \[ = - 5 \tan{\left(3 x - 3 \right)} + \frac{5 \cot{\left(3 x - 3 \right)}}{\cos^{2}{\left(3 x - 3 \right)}} \]algebraRewrite secant squared as 1/cos squared.≈ Checked numerically
- \[ = - 5 \tan{\left(3 x - 3 \right)} + \frac{5}{\sin{\left(3 x - 3 \right)} \cos{\left(3 x - 3 \right)}} \]algebra algebra algebraRewrite cotangent in terms of sine and cosine. Simplify the fraction by canceling one cosine term. Multiply numerator and denominator by 2 to prepare for double angle identity.✓ Proved
- \[ = - 5 \tan{\left(3 x - 3 \right)} + \frac{10}{\sin{\left(6 x - 6 \right)}} \]algebraApply the double angle identity for sine.✓ Proved
- \[ = - 5 \tan{\left(3 x - 3 \right)} + 10 \csc{\left(6 x - 6 \right)} \]simplifyFinal simplification using the cosecant identity.✓ Proved
Answer \( \frac{5}{\tan{\left(3 x - 3 \right)}} \)
Lines: 15 proved, 2 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 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 cot has poles at multiples of pi |
| 11 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 5*((tan(3*x - 3) + 2/sin(6*x - 6))*cos(3*x - 3)**2 - sin(6*x - 6)/2 - cot(3*x - 3))/cos(3*x - 3)**2; numeric agreement only, at 24 of 24 sampled points sec has poles at odd multiples of pi/2 cot has poles at multiples of pi undefined where cos(3*x - 3) = 0 tan has poles at odd multiples of pi/2 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 cot has poles at multiples of pi undefined where cos(3*x - 3) = 0 undefined where sin(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 |
| 15 | ✓ 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 |
| 16 | ✓ 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 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -5*tan(3*x - 3) - 5/tan(3*x - 3) + 10/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 the given function is not 5/tan(3*x-3); the intermediate algebraic steps lead to a different expression, and the final simplification to 10*csc(6*x-6) is also not equivalent to the stated answer.qwen3.6:27b-mlx: fail (error) — The final answer provided (5/tan(3*x - 3)) does not match the result of the derivation steps (which simplify to 10*csc(6*x - 6) - 5*tan(3*x - 3)). The stated answer is mathematically incorrect for the given function.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The final answer provided (5/tan(3*x - 3)) does not match the result of the derivation steps (which simplify to 10*csc(6*x - 6) - 5*tan(3*x - 3)). The stated answer is mathematically incorrect for the given function.gpt-oss:20b: fail (error) 2026-09-27 — The final derivative is incorrect. The correct derivative of the given function is not 5/tan(3*x-3); the intermediate algebraic steps lead to a different expression, and the final simplification to 10*csc(6*x-6) is also not equivalent to the stated answer.qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final result is mathematically incorrect; the derivative of the given function is 0, not 5/tan(3*x - 3). The solution fails to recognize that thgpt-oss:20b: fail (error) 2026-09-27 — Step 5 applies both the power rule and the chain rule in a single line, violating the rule that each step must change only one thing. This multi‑rule application is a defect.
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.