Derivative of \( \displaystyle - \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \)
Problem 2.415 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \]constantPull out the constant -1.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}\right)}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(x + 1 \right)} + \frac{d}{d x} \csc{\left(x + 1 \right)}}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]sumApply the sum rule to the inner terms.✓ Proved
- \[ = - \frac{- \cot{\left(x + 1 \right)} \csc{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) - \csc^{2}{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right)}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]trigDifferentiate the trigonometric functions.✓ Proved
- \[ = - \frac{- \cot{\left(x + 1 \right)} \csc{\left(x + 1 \right)} - \csc^{2}{\left(x + 1 \right)}}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]chainDifferentiate the inner linear function x + 1.✓ Proved
- \[ = \csc{\left(x + 1 \right)} \]algebra algebra simplifyFactor out -csc(x + 1) from the numerator. Cancel the common term in the numerator and denominator. Simplify the remaining expression.✓ Proved
Answer \( \frac{1}{\sin{\left(x + 1 \right)}} \)
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 Every line of this solution was proved by the computer algebra system SymPy. 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(x + 1) + csc(x + 1) = 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(x + 1) + csc(x + 1) = 0 |
| 5 | ✓ 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(x + 1) + csc(x + 1) = 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(x + 1) + csc(x + 1) = 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(x + 1) + csc(x + 1) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi |
| 9 | ✓ 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(x + 1) = 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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately describe the operations performed.gpt-oss:20b: pass 2026-09-20
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.