Derivative of \( \displaystyle - \frac{5 \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{5 \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \)
Problem 2.2441 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{5 \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{5 \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2}\right) \]derivative constant-multipleStart with the derivative of the function. Factor out the constants.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4}\right) + \frac{d}{d x} \frac{5 \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{5 \frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]constant-multipleFactor out the constants from each term.✓ Proved
- \[ = - \frac{5 \cos^{2}{\left(2 x + 2 \right)} \frac{d}{d x} \frac{1}{\cos^{2}{\left(2 x + 2 \right)}}}{4} + \frac{5 \frac{d}{d x} \tan{\left(2 x + 2 \right)}}{2 \tan{\left(2 x + 2 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = - \frac{5 \cos^{2}{\left(2 x + 2 \right)} \frac{d}{d x} \frac{1}{\cos^{2}{\left(2 x + 2 \right)}}}{4} + \frac{5 \sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]chainDifferentiate the tangent term.✓ Proved
- \[ = \frac{5 \sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} + \frac{5 \frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 \cos{\left(2 x + 2 \right)}} \]chainApply the chain rule to the power term.✓ Proved
- \[ = - \frac{5 \sin{\left(2 x + 2 \right)}}{\cos{\left(2 x + 2 \right)}} + \frac{5 \sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]chain algebra algebra algebraDifferentiate the cosine term. Simplify the numerator of the first term. Simplify the powers of cosine. Simplify the coefficients and fractions.✓ Proved
- \[ = - 5 \tan{\left(2 x + 2 \right)} + \frac{5 \sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]algebraRewrite the first term using tangent.✓ Proved
- \[ = - 5 \tan{\left(2 x + 2 \right)} + \frac{5}{\sin{\left(2 x + 2 \right)} \cos{\left(2 x + 2 \right)}} \]trig algebra algebraRewrite secant in terms of cosine and tangent. Simplify the fraction. Prepare for the double angle identity.✓ Proved
- \[ = - 5 \tan{\left(2 x + 2 \right)} + \frac{10}{\sin{\left(4 x + 4 \right)}} \]algebraApply the double angle identity sin(2u) = 2sin(u)cos(u).✓ Proved
- \[ = - 5 \tan{\left(2 x + 2 \right)} + 10 \csc{\left(4 x + 4 \right)} \]trigRewrite 1/sin as csc.✓ Proved
- \[ = - 5 \tan{\left(2 x + 2 \right)} + \frac{10}{\sin{\left(4 x + 4 \right)}} \]simplifyFinal simplified form.✓ Proved
Answer \( \frac{5}{\tan{\left(2 x + 2 \right)}} \)
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: 18 proved, 1 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. How we verify
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(2*x + 2) = 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(2*x + 2) = 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(2*x + 2) = 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 undefined where cos(2*x + 2) = 0 undefined where tan(2*x + 2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(2*x + 2) = 0 undefined where tan(2*x + 2) = 0 sec has poles at odd multiples of pi/2 |
| 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 cos(2*x + 2) = 0 undefined where tan(2*x + 2) = 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 cos(2*x + 2) = 0 undefined where tan(2*x + 2) = 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 cos(2*x + 2) = 0 undefined where tan(2*x + 2) = 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 cos(2*x + 2) = 0 undefined where tan(2*x + 2) = 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 cos(2*x + 2) = 0 undefined where tan(2*x + 2) = 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 cos(2*x + 2) = 0 undefined where tan(2*x + 2) = 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(2*x + 2) = 0 undefined where cos(2*x + 2) = 0 undefined where sin(2*x + 2) = 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(2*x + 2) = 0 undefined where sin(2*x + 2) = 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(2*x + 2) = 0 undefined where sin(2*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 cos(2*x + 2) = 0 undefined where sin(2*x + 2) = 0 undefined where sin(4*x + 4) = 0 |
| 17 | ✓ 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 + 4) = 0 csc has poles at multiples of pi |
| 18 | ✓ 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(4*x + 4) = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -5/(cos(2*x + 2)**2*tan(2*x + 2)) + 10/sin(4*x + 4); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(2*x + 2) = 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 simplified derivative is incorrect. The correct derivative simplifies to 5/(sin(4*x+4)), not 5/tan(2*x+2).qwen3.6:27b-mlx: fail (error) — The final result does not match the stated answer. The simplification of the first term in step 11 is incorrect: -5/4 * 4 * sin/cos equals -5*tan, but the derivative of the first term of the original function simplifies to 5*tan(2x+2), not -5*tan(2x+2). Specifically, log(cos(u)^(-2)) = -2*log(cos(u)), so its derivative is -2 * (-sin(u)/cos(u)) * u' = 2*tan(u)*u'. With the coefficient -5/4 and u'=2, this is (-5/4)*2*2*tan = -5*tan? Wait. Let's re-evaluate. f1 = -5/4 * log(cos(2x+2)^(-2)) = -5/4 * (-2) * log(cos(2x+2)) = 5/2 * log(cos(2x+2)). Derivative is 5/2 * (-sin(2x+2)/cos(2x+2)) * 2 = -5 * tan(2x+2). So the first term is indeed -5*tan(2x+2). The second term is 5/2 * log(tan(2x+2)). Derivative is 5/2 * (sec^2(2x+2)/tan(2x+2)) * 2 = 5 * sec^2(2x+2)/tan(2x+2). So the sum is -5*tan(2x+2) + 5*sec^2(2x+2)/tan(2x+2). This simplifies to -5*sin/cos + 5/(cos^2 * cos/sin) = -5*sin/cos + 5*sin/cos^3? No. 5*sec^2/tan = 5/(cos^2 * sin/cos) = 5/(sin*cos). So the result is -5*tan(2x+2) + 5/(sin(2x+2)cos(2x+2)). The stated answer is 5/tan(2x+2) = 5*cot(2x+2). Is -5*tan + 5/(sin*cos) equal to 5*cot? -5*sin/cos + 5/(sin*cos) = (-5*sin^2 + 5)/(sin*cos) = 5*(1-sin^2)/(sin*cos) = 5*cos^2/(sin*cos) = 5*cos/sin = 5*cot. Yes, the math is correct. The stated answer is 5/tan(2x+2). The solution ends at step 18 with -5*tan(2x+2) + 10/sin(4x+4). This is equivalent to 5*cot(2x+2). However, the solution does not explicitly reach the stated answer form. It stops at an equivalent but different form. Is this a defect? The prompt says 'Stated answer: 5/tan(2*x + 2)'. The solution does not produce this exact form. It produces an equivalent form. Usually, 'pass' requires reaching the stated answer or an explicitly equivalent final form if the steps are correct. But wait, look at step 18. It says 'Final simplified form'. It is not the stated answer. Is it a 'style' issue or 'error'? If the task is to differentiate and the answer is given, the solution should ideally match. However, often equivalent forms are accepted. Let's look closer. Step 18 is just a rewrite of step 16. Step 17 was csc, step 18 is 1/sin. This is redundant. But the main issue is that the final expression is not the stated answer. Is it 'misleading'? No, it's correct. Is it 'style'? Maybe. But let's look at the steps again. Step 1 to 12 are correct. Step 12: -5*tan + 5*sec^2/tan. Step 13: rewrite sec and tan. Step 14: simplify. Step 15: prepare for double angle. Step 16: apply double angle. Step 17: rewrite as csc. Step 18: rewrite back to 1/sin. The solution never combines the terms to get 5*cot. It leaves them separate. The stated answer is a single term. The solution is two terms. This is a failure to simplify to the stated answer. Is that a 'style' defect? Or 'error'? The instructions say 'find what is wrong'. If the answer is mathematically equivalent, it's usually a style issue unless the contract demands the specific form. The contract says 'Stated answer: ...'. It doesn't explicitly say 'must match exactly'. However, in automated grading, equivalence is key. But here, I am reviewing the *solution steps*. The steps are correct. The final result is equivalent. I will mark it as pass because the mathematics is sound and the final result is equivalent to the stated answer. Wait, let me re-read the prompt. 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. It does not confirm the final line equals the stated answer. But usually, if the final line is equivalent, it's a pass. Let's look for a real defect. Step 5: 'chain' rule for log. Correct. Step 6: 'chain' for tan. Correct. Step 7: 'chain' for power. Correct. Step 8: 'chain' for cos. Correct. Step 9: 'algebra'. Correct. Step 10: 'algebra'. Correct. Step 11: 'algebra'. Correct. Step 12: 'algebra'. Correct. Step 13: 'trig'. Correct. Step 14: 'algebra'. Correct. Step 15: 'algebra'. Correct. Step 16: 'algebra'. Correct. Step 17: 'trig'. Correct. Step 18: 'simplify'. Correct. The solution is correct. The final form is equivalent. I will pass it.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The final result does not match the stated answer. The simplification of the first term in step 11 is incorrect: -5/4 * 4 * sin/cos equals -5*tan, but the derivative of the first term of the original function simplifies to 5*tan(2x+2), not -5*tan(2x+2). Specifically, log(cos(u)^(-2)) = -2*log(cos(u)), so its derivative is -2 * (-sin(u)/cos(u)) * u' = 2*tan(u)*u'. With the coefficient -5/4 and u'=2, this is (-5/4)*2*2*tan = -5*tan? Wait. Let's re-evaluate. f1 = -5/4 * log(cos(2x+2)^(-2)) = -5/4 * (-2) * log(cos(2x+2)) = 5/2 * log(cos(2x+2)). Derivative is 5/2 * (-sin(2x+2)/cos(2x+2)) * 2 = -5 * tan(2x+2). So the first term is indeed -5*tan(2x+2). The second term is 5/2 * log(tan(2x+2)). Derivative is 5/2 * (sec^2(2x+2)/tan(2x+2)) * 2 = 5 * sec^2(2x+2)/tan(2x+2). So the sum is -5*tan(2x+2) + 5*sec^2(2x+2)/tan(2x+2). This simplifies to -5*sin/cos + 5/(cos^2 * cos/sin) = -5*sin/cos + 5*sin/cos^3? No. 5*sec^2/tan = 5/(cos^2 * sin/cos) = 5/(sin*cos). So the result is -5*tan(2x+2) + 5/(sin(2x+2)cos(2x+2)). The stated answer is 5/tan(2x+2) = 5*cot(2x+2). Is -5*tan + 5/(sin*cos) equal to 5*cot? -5*sin/cos + 5/(sin*cos) = (-5*sin^2 + 5)/(sin*cos) = 5*(1-sin^2)/(sin*cos) = 5*cos^2/(sin*cos) = 5*cos/sin = 5*cot. Yes, the math is correct. The stated answer is 5/tan(2x+2). The solution ends at step 18 with -5*tan(2x+2) + 10/sin(4x+4). This is equivalent to 5*cot(2x+2). However, the solution does not explicitly reach the stated answer form. It stops at an equivalent but different form. Is this a defect? The prompt says 'Stated answer: 5/tan(2*x + 2)'. The solution does not produce this exact form. It produces an equivalent form. Usually, 'pass' requires reaching the stated answer or an explicitly equivalent final form if the steps are correct. But wait, look at step 18. It says 'Final simplified form'. It is not the stated answer. Is it a 'style' issue or 'error'? If the task is to differentiate and the answer is given, the solution should ideally match. However, often equivalent forms are accepted. Let's look closer. Step 18 is just a rewrite of step 16. Step 17 was csc, step 18 is 1/sin. This is redundant. But the main issue is that the final expression is not the stated answer. Is it 'misleading'? No, it's correct. Is it 'style'? Maybe. But let's look at the steps again. Step 1 to 12 are correct. Step 12: -5*tan + 5*sec^2/tan. Step 13: rewrite sec and tan. Step 14: simplify. Step 15: prepare for double angle. Step 16: apply double angle. Step 17: rewrite as csc. Step 18: rewrite back to 1/sin. The solution never combines the terms to get 5*cot. It leaves them separate. The stated answer is a single term. The solution is two terms. This is a failure to simplify to the stated answer. Is that a 'style' defect? Or 'error'? The instructions say 'find what is wrong'. If the answer is mathematically equivalent, it's usually a style issue unless the contract demands the specific form. The contract says 'Stated answer: ...'. It doesn't explicitly say 'must match exactly'. However, in automated grading, equivalence is key. But here, I am reviewing the *solution steps*. The steps are correct. The final result is equivalent. I will mark it as pass because the mathematics is sound and the final result is equivalent to the stated answer. Wait, let me re-read the prompt. 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. It does not confirm the final line equals the stated answer. But usually, if the final line is equivalent, it's a pass. Let's look for a real defect. Step 5: 'chain' rule for log. Correct. Step 6: 'chain' for tan. Correct. Step 7: 'chain' for power. Correct. Step 8: 'chain' for cos. Correct. Step 9: 'algebra'. Correct. Step 10: 'algebra'. Correct. Step 11: 'algebra'. Correct. Step 12: 'algebra'. Correct. Step 13: 'trig'. Correct. Step 14: 'algebra'. Correct. Step 15: 'algebra'. Correct. Step 16: 'algebra'. Correct. Step 17: 'trig'. Correct. Step 18: 'simplify'. Correct. The solution is correct. The final form is equivalent. I will pass it.gpt-oss:20b: fail (error) 2026-10-19 — The final simplified derivative is incorrect. The correct derivative simplifies to 5/(sin(4*x+4)), not 5/tan(2*x+2).qwen3.6:27b-mlx: fail (error) 2026-10-17 — The final result does not match the stated answer (5/tan(2*x + 2)). The derivation incorrectly simplifies the first term to -5*tan(2*x+2) instead of +5*tan(2*x+2), leading to a non-zero result rather than the correct cancellation.gpt-oss:20b: inconclusive 2026-10-17 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final simplification is incorrect: the derivative simplifies to \u22155\tan(2x+2)^{-1} = 5\cos(2x+2)/\sin(2x+2), not to the expression obtained in theqwen3.6:27b-mlx: fail (error) 2026-10-17 — The final answer provided in the task description (5/tan(2*x + 2)) does not match the result of the derivation (-5*tan(2*x + 2) + 10/sin(4*x + 4)). The derivation is mathematically correct, but the solution fails to simplify to the stated answer, indicating a mismatch between the steps and the claimed result.gpt-oss:20b: fail (error) 2026-10-17 — Step 17 incorrectly rewrites 5/(sin(2x+2)cos(2x+2)) as 10 csc(4x+4). The correct simplification is 5 csc(2x+2) sec(2x+2) (or 5/(sin cos)), not 10 csc. This makes the final derivative incorrect.
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 2026-10-19 with SymPy 1.14.0.