Derivative of \( \displaystyle - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{3} + \frac{2 \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \)
Problem 2.1498 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{3} + \frac{2 \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{3} + \frac{2 \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3}\right) \]derivative rewriteDifferentiate the function. Rewrite the power using the exponential identity.✓ Proved
- \[ = \frac{d}{d x} \left(\frac{2 \ln{\left(\cos{\left(3 x - 3 \right)} \right)}}{3} + \frac{2 \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3}\right) \]simplify algebra constant-multipleSimplify the nested logarithm. Simplify the signs and coefficients. Factor out the common constant.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \left(\ln{\left(\cos{\left(3 x - 3 \right)} \right)} + \ln{\left(\tan{\left(3 x - 3 \right)} \right)}\right)}{3} \]constant-multipleMove the constant outside the derivative.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)}}{3} + \frac{2 \frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \]sumApply the sum rule.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \tan{\left(3 x - 3 \right)}}{3 \tan{\left(3 x - 3 \right)}} + \frac{2 \frac{d}{d x} \cos{\left(3 x - 3 \right)}}{3 \cos{\left(3 x - 3 \right)}} \]chainApply the chain rule to both terms.✓ Proved
- \[ = - \frac{2 \sin{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right)}{3 \cos{\left(3 x - 3 \right)}} + \frac{2 \sec^{2}{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right)}{3 \tan{\left(3 x - 3 \right)}} \]chainApply the chain rule to the inner functions.✓ Proved
- \[ = - 2 \tan{\left(3 x - 3 \right)} + \frac{2 \sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]derivative constant-multiple algebraDifferentiate the inner linear functions. Simplify the coefficients. Distribute the 1/3 and 3.✓ Proved
- \[ = - 2 \tan{\left(3 x - 3 \right)} + \frac{2}{\sin{\left(3 x - 3 \right)} \cos{\left(3 x - 3 \right)}} \]rewrite algebraRewrite 1/tan as cot. Simplify the product of trigonometric terms.✓ Proved
- \[ = - 2 \tan{\left(3 x - 3 \right)} + \frac{4}{\sin{\left(6 x - 6 \right)}} \]algebraUse the double angle identity for sine.✓ Proved
- \[ = - 2 \tan{\left(3 x - 3 \right)} + 4 \csc{\left(6 x - 6 \right)} \]rewrite simplifyRewrite 1/sin as csc. Distribute the 2 and finalize the expression.✓ Proved
Answer \( \frac{2}{\tan{\left(3 x - 3 \right)}} \)
Lines: 17 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 |
| 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 |
| 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 |
| 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 undefined where cos(3*x - 3) = 0 undefined where tan(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 undefined where cos(3*x - 3) = 0 undefined where tan(3*x - 3) = 0 sec has poles at odd multiples of pi/2 |
| 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 cos(3*x - 3) = 0 undefined where tan(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 |
| 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(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 sec has poles at odd multiples of pi/2 undefined where tan(3*x - 3) = 0 undefined where cos(3*x - 3) = 0 undefined where sin(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 cos(3*x - 3) = 0 undefined where sin(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 cos(3*x - 3) = 0 undefined where sin(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 |
| 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 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -2*tan(3*x - 3) - 2/tan(3*x - 3) + 4/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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final derivative is incorrectly simplified to \u201c-2\tan(3x-3)+4\csc(6x-6)\u201d, which does not equal the correct derivative \u201c2/\tan(3x-3)\u20qwen3.6:27b-mlx: fail (error) — The final answer is mathematically incorrect; the derivative of the given function is 2*cot(3*x - 3), not 2/tan(3*x - 3) (which is 2*cot(3*x - 3) only if one ignores the sign error in the first term's simplification or the fact that the two terms in the original function do not cancel to produce the stated result). Specifically, step 10 simplifies the first term to -tan(...) but the original function's first term -log(cos(...)^-2)/3 simplifies to (2/3)log(cos(...)), whose derivative is -2/3 * tan(...). The second term 2/3 log(tan(...)) has derivative 2/3 * sec^2/tan = 2/3 * 1/sin*cos = 2/3 * 2/sin(2u). The sum is not 2/tan. The solver's final expression -2*tan + 4*csc is not equal to 2/tan. The stated answer 2/tan(3*x-3) is wrong.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The final answer is mathematically incorrect; the derivative of the given function is 2*cot(3*x - 3), not 2/tan(3*x - 3) (which is 2*cot(3*x - 3) only if one ignores the sign error in the first term's simplification or the fact that the two terms in the original function do not cancel to produce the stated result). Specifically, step 10 simplifies the first term to -tan(...) but the original function's first term -log(cos(...)^-2)/3 simplifies to (2/3)log(cos(...)), whose derivative is -2/3 * tan(...). The second term 2/3 log(tan(...)) has derivative 2/3 * sec^2/tan = 2/3 * 1/sin*cos = 2/3 * 2/sin(2u). The sum is not 2/tan. The solver's final expression -2*tan + 4*csc is not equal to 2/tan. The stated answer 2/tan(3*x-3) is wrong.gpt-oss:20b: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final derivative is incorrectly simplified to \u201c-2\tan(3x-3)+4\csc(6x-6)\u201d, which does not equal the correct derivative \u201c2/\tan(3x-3)\u20qwen3.6:27b-mlx: fail (error) 2026-10-04 — The final result is mathematically incorrect; the derivative of the given function is 0 (since the terms cancel out), but the solution yields a non-zero expression. This is likely due to an error in Step 9 where the derivative of cos is taken as -sin without the chain rule factor for the inner function being applied correctly to the first term, or more likely, the simplification in Step 10/11 incorrectly handles the cancellation. Specifically, d/dx(log(cos(u))) = -tan(u)*u' and d/dx(log(tan(u))) = 2*csc(2u)*u'. With u=3x-3, u'=3. The terms are -tan(u)*3 and 2*csc(2u)*3. These do not cancel to 0 generally. Wait, let's re-evaluate. f(x) = -log(cos(u)^-2)/3 + 2log(tan(u))/3 = 2log(cos(u))/3 + 2log(tan(u))/3 = 2/3 log(cos(u)tan(u)) = 2/3 log(sin(u)). Derivative is 2/3 cot(u) * 3 = 2 cot(u). The stated answer is 2/tan(u) = 2 cot(u). So the stated answer is correct. Let's re-read the steps. Step 8: 2/3 * ( (1/cos)*(-sin)*3 + (1/tan)*(sec^2)*3 ). Step 10: 2/3 * ( -tan*3 + (1/tan)*(sec^2)*3 ). Step 11: 2 * ( -tan + (1/tan)*(sec^2) ). Step 12: 2 * ( -tan + cot*sec^2 ). cot*sec^2 = (cos/sin)*(1/cos^2) = 1/(sin*cos) = 2/sin(2u). So result is 2(-tan(u) + 2/sin(2u)). Is -tan(u) + 2/sin(2u) equal to cot(u)? -sin/cos + 2/(2sin cos) = -sin/cos + 1/(sin cos) = (-sin^2 + 1)/(sin cos) = cos^2/(sin cos) = cos/sin = cot. Yes. So the math is correct. Why did I think it was 0? log(cos^-2) = -2 log cos. So -(-2 log cos)/3 = 2/3 log cos. 2/3 log tan. Sum = 2/3 log(cos tan) = 2/3 log(sin). Deriv = 2/3 cot * 3 = 2 cot. The solution arrives at 2(-tan + 2/sin(2u)) which simplifies to 2 cot. The steps are algebraically correct. Let's look for rule violations. Step 8 applies chain rule to BOTH terms. The rule is 'chain'. Is applying a rule to two terms at once a defect? 'Each step changes ONE thing. A step that applies two rules at once is a defect.' Step 8 applies the chain rule to the first term AND the chain rule to the second term. This is applying the same rule to two different sub-expressions simultaneously. This violates the 'one thing' constraint. Step 9 also applies chain rule to two terms. Step 10 applies derivative to two terms. These are compound steps.gpt-oss:20b: fail (style) 2026-10-04 — Step 13 applies two rewrites (1/tan to cot and sec^2 to 1/cos^2) in a single step, violating the one‑rule‑per‑step rule.
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 2026-10-04 with SymPy 1.14.0.