Derivative of \( \displaystyle - \frac{\ln{\left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)} \right)}}{2} \)
Problem 2.1496 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)} \right)}}{2}\right) \]constantStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)} \right)}}{2} \]constantPull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)}\right)}{2 \left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)}\right)} \]logarithmicApply the chain rule for the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(4 x + 2 \right)} + \frac{d}{d x} \csc{\left(4 x + 2 \right)}}{2 \left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)}\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = - \frac{- \cot{\left(4 x + 2 \right)} \csc{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right) - \csc^{2}{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right)}{2 \left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)}\right)} \]trigApply the derivative rules for cotangent and cosecant.≈ Checked numerically
- \[ = - \frac{- 4 \cot{\left(4 x + 2 \right)} \csc{\left(4 x + 2 \right)} - 4 \csc^{2}{\left(4 x + 2 \right)}}{2 \left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)}\right)} \]chain constant-multipleApply the chain rule to the inner linear function. Factor out the common constant 4.✓ Proved
- \[ = \frac{2 \cot{\left(4 x + 2 \right)} \csc{\left(4 x + 2 \right)} + 2 \csc^{2}{\left(4 x + 2 \right)}}{\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)}} \]algebraSimplify the signs and the constant multiplication.✓ Proved
- \[ = 2 \csc{\left(4 x + 2 \right)} \]algebra simplifyFactor out csc(4*x + 2) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{2}{\sin{\left(4 x + 2 \right)}} \)
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.
✓ Nihil obstat Lines: 10 proved, 1 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 csc has poles at multiples of pi cot has poles at multiples of pi |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments csc has poles at multiples of pi cot has poles at multiples of pi undefined where cot(4*x + 2) + csc(4*x + 2) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where cot(4*x + 2) + csc(4*x + 2) = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*(-csc(4*x + 2)**2 + 1 + tan(4*x + 2)**(-2))*sin(4*x + 2)/(cos(4*x + 2) + 1); numeric agreement only, at 24 of 24 sampled points csc has poles at multiples of pi cot has poles at multiples of pi undefined where cot(4*x + 2) + csc(4*x + 2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where cot(4*x + 2) + csc(4*x + 2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where cot(4*x + 2) + csc(4*x + 2) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where cot(4*x + 2) + csc(4*x + 2) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where cot(4*x + 2) + csc(4*x + 2) = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(4*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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is algebraically sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is algebraically sound.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04
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.