Derivative of \( \displaystyle \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} - \frac{\ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \)
Problem 2.2460 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} - \frac{\ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(\frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} - \frac{\ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2}\right) \]sumApply the sum rule for derivatives.✓ Proved
- \[ = \frac{d}{d x} \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} - \frac{d}{d x} \frac{\ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]constant-multipleFactor out the constants.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} - \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]rewriteRewrite the power using the exponential-logarithm identity.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- 2 \ln{\left(\cos{\left(2 x + 2 \right)} \right)}\right)}{4} - \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]algebraSimplify the exponent.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(2 x + 2 \right)} \right)}}{2} - \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]constant-multiple constant-multipleDistribute the constant 1/4 * (-2). Factor out -1/2.✓ Proved
- \[ = - \frac{\frac{d}{d x} \tan{\left(2 x + 2 \right)}}{2 \tan{\left(2 x + 2 \right)}} - \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 \cos{\left(2 x + 2 \right)}} \]chainApply the chain rule to each term.✓ Proved
- \[ = \frac{\sin{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{2 \cos{\left(2 x + 2 \right)}} - \frac{\sec^{2}{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{2 \tan{\left(2 x + 2 \right)}} \]derivativeDifferentiate the inner functions.✓ Proved
- \[ = \frac{\sin{\left(2 x + 2 \right)}}{\cos{\left(2 x + 2 \right)}} - \frac{\sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]algebraEvaluate the derivative of (2*x + 2).✓ Proved
- \[ = \tan{\left(2 x + 2 \right)} - \frac{\sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]algebra algebraSimplify the expression inside the parentheses. Distribute -1/2.✓ Proved
- \[ = \tan{\left(2 x + 2 \right)} - \frac{1}{\sin{\left(2 x + 2 \right)} \cos{\left(2 x + 2 \right)}} \]algebra algebra algebraRewrite sec(2*x + 2)**2 as 1/cos(2*x + 2)**2. Simplify the complex fraction. Introduce a 2 in the numerator and denominator.✓ Proved
- \[ = \tan{\left(2 x + 2 \right)} - \frac{2}{\sin{\left(4 x + 4 \right)}} \]algebraUse the double angle identity sin(2u) = 2*sin(u)*cos(u).✓ Proved
- \[ = \tan{\left(2 x + 2 \right)} - 2 \csc{\left(4 x + 4 \right)} \]simplifyRewrite 1/sin(u) as csc(u).✓ Proved
Answer \( - \frac{1}{\tan{\left(2 x + 2 \right)}} \)
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: 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. 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 |
| 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 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 undefined where cos(2*x + 2) = 0 undefined where tan(2*x + 2) = 0 sec has poles at odd multiples of pi/2 |
| 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 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 tan(2*x + 2) = 0 undefined where cos(2*x + 2) = 0 undefined where sin(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 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 undefined where sin(4*x + 4) = 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(4*x + 4) = 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(2*x + 2) + 1/tan(2*x + 2) - 2/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 derivative tan(2*x+2) - 2*csc(4*x+4) is incorrect; the correct derivative is tan(2*x+2) - 1/(sin(2*x+2)*cos(2*x+2)), which does not simplify to -1/tan(2*x+2).qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 3 incorrectly rewrites log(cos(2*x + 2)**(-2)) as log(exp(-2 * log(cos(2*x + 2)))). This identity log(u^v) = v*log(u) is not generally valid (i
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 3 incorrectly rewrites log(cos(2*x + 2)**(-2)) as log(exp(-2 * log(cos(2*x + 2)))). This identity log(u^v) = v*log(u) is not generally valid (igpt-oss:20b: fail (error) 2026-10-19 — The final derivative tan(2*x+2) - 2*csc(4*x+4) is incorrect; the correct derivative is tan(2*x+2) - 1/(sin(2*x+2)*cos(2*x+2)), which does not simplify to -1/tan(2*x+2).qwen3.6:27b-mlx: fail (error) 2026-10-18 — Step 1 is labeled 'sum' but does not perform the sum rule; it merely restates the problem. Step 2 is labeled 'constant-multiple' but performs the sum rule (splitting the derivative of a difference). The labels are swapped and incorrect for the operations shown.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-18 — The final answer is mathematically incorrect; the derivative simplifies to 0, not -1/tan(2*x + 2). The error occurs in Step 11, where the constant factor -1/2 is dropped instead of being distributed, leading to an incorrect result.gpt-oss:20b: fail (error) 2026-10-18 — The final result tan(2*x+2)-2*csc(4*x+4) does not equal the correct derivative -1/tan(2*x+2). The solution contains a fundamental algebraic error in simplifying the expression, leading to an incorrect final answer.
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.