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 2.277 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2}\right) \]sumStart with the derivative of the sum.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4}\right) + \frac{d}{d x} \frac{\ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]constant-multipleSeparate the terms and pull out constants.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]logarithmicApply the chain rule for the logarithm: d/dx log(u) = (1/u) * du/dx.✓ Proved
- \[ = - \frac{\cos^{2}{\left(2 x + 2 \right)} \frac{d}{d x} \frac{1}{\cos^{2}{\left(2 x + 2 \right)}}}{4} + \frac{\frac{d}{d x} \tan{\left(2 x + 2 \right)}}{2 \tan{\left(2 x + 2 \right)}} \]chainApply the chain rule to the inner power function.✓ 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 the cosine function.✓ Proved
- \[ = - \frac{\sin{\left(2 x + 2 \right)}}{\cos{\left(2 x + 2 \right)}} + \frac{\sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]chain algebra algebra algebra algebraApply the chain rule to the innermost argument (2x + 2). Simplify the product of terms in the first part. Simplify the fraction involving powers of cosine. Simplify the coefficient of the first term. Rewrite the power term as a fraction.✓ Proved
- \[ = - \tan{\left(2 x + 2 \right)} + \frac{\sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]simplifyRecognize the definition of tangent.✓ Proved
- \[ = - \tan{\left(2 x + 2 \right)} + \frac{1}{\sin{\left(2 x + 2 \right)} \cos{\left(2 x + 2 \right)}} \]rewrite algebra algebraRewrite secant in terms of cosine. Simplify the second term. Prepare for the double angle identity.✓ Proved
- \[ = - \tan{\left(2 x + 2 \right)} + \frac{2}{\sin{\left(4 x + 4 \right)}} \]algebraApply the sine double angle identity.✓ Proved
- \[ = - \tan{\left(2 x + 2 \right)} + 2 \csc{\left(4 x + 4 \right)} \]rewriteFinal simplification using the cosecant identity.✓ 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.
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 undefined where tan(2*x + 2) = 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(2*x + 2) = 0 undefined where cos(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 tan(2*x + 2) = 0 undefined where cos(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 tan(2*x + 2) = 0 undefined where cos(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 tan(2*x + 2) = 0 undefined where cos(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 tan(2*x + 2) = 0 undefined where cos(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 tan(2*x + 2) = 0 undefined where cos(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 undefined where cos(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 result does not match the correct derivative 1/tan(2*x+2); the simplification steps from line 10 onward introduce an incorrect extra term. The derivative of the given function is 1/tan(2*x+2), not the expression obtained in the solution.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 2 applies two rules at once (linearity of the derivative to split the sum, and the constant multiple rule to pull out coefficients), violating the one-rule-per-step constraint. Additionally, Step 3 is labeled 'logarithmic' but performs no differentiation; the actual application of the logarithmic derivative rule occurs in Step 4.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 applies two rules at once (linearity of the derivative to split the sum, and the constant multiple rule to pull out coefficients), violating the one-rule-per-step constraint. Additionally, Step 3 is labeled 'logarithmic' but performs no differentiation; the actual application of the logarithmic derivative rule occurs in Step 4.gpt-oss:20b: fail (error) 2026-09-20 — The final result does not match the correct derivative 1/tan(2*x+2); the simplification steps from line 10 onward introduce an incorrect extra term. The derivative of the given function is 1/tan(2*x+2), not the expression obtained in the solution.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 applies two rules at once (linearity of the derivative to split the sum, and constant multiple rule to pull out coefficients), violating the one-rule-per-step constraint. Additionally, Step 3 is labeled 'logarithmic' but performs no differentiation; it merely rewrites the constant factors, making the label incorrect for the transformation shown.gpt-oss:20b: fail (error) 2026-09-20 — The final derivative is incorrectly simplified; the correct result is 1/tan(2*x+2), not -tan(2*x+2)+2*csc(4*x+4).qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 applies both the sum rule (splitting the derivative) and the constant multiple rule (pulling out constants) simultaneously, violating the one-rule-per-step constraint. Additionally, Step 3 is labeled 'logarithmic' but performs no differentiation; it merely prepares for the chain rule application in Step 4, making the label incorrect for the transformation shown.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 3 incorrectly labels the application of the chain rule for the logarithm as "logarithmic"; it should be labeled "chain". This mislabeling violates the rule‑granularity requirement.qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 applies the sum rule to split the derivative and the constant multiple rule to pull out constants simultaneously, violating the one-rule-per-step constraint. Additionally, the final answer provided (1/tan(2*x + 2)) does not match the simplified result of the steps (-tan(2*x + 2) + 2*csc(4*x + 4)), indicating a logical disconnect between the derivation and the stated answer.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: inconclusive 2026-09-17 — reviewer response could not be parsed:deepseek-r1:70b: pass 2026-09-17
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 2026-09-26 with SymPy 1.14.0.