Derivative of \( \displaystyle - \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)} \)
Problem 2.247 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)} \]constantPull the constant -1 out of the derivative.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}\right)}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]chainApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(x + 2 \right)} + \frac{d}{d x} \csc{\left(x + 2 \right)}}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]sumApply the sum rule to the inner expression.✓ Proved
- \[ = - \frac{- \cot{\left(x + 2 \right)} \csc{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right) - \csc^{2}{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right)}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]trigDifferentiate the trigonometric functions using the chain rule.✓ Proved
- \[ = - \frac{- \cot{\left(x + 2 \right)} \csc{\left(x + 2 \right)} - \csc^{2}{\left(x + 2 \right)}}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]derivativeEvaluate the derivatives of the inner linear terms.✓ Proved
- \[ = \csc{\left(x + 2 \right)} \]algebra algebra simplifyFactor out the common term -csc(x + 2). Cancel the common term in the numerator and denominator. Simplify the remaining expression.✓ Proved
Answer \( \frac{1}{\sin{\left(x + 2 \right)}} \)
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.
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. 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 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 + 2) + csc(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(x + 2) + csc(x + 2) = 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 + 2) + csc(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(x + 2) + csc(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(x + 2) + csc(x + 2) = 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 + 2) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives in distinct steps. The algebraic simplification is sound, and all labels correspond to the rules applied.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives in distinct steps. The algebraic simplification is sound, and all labels correspond to the rules applied.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives in distinct steps. The algebraic simplification and cancellation are handled properly with appropriate labels.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies both the trigonometric derivative rule and the chain rule in one line, violating the one‑rule‑per‑step rule. It should be split into two separate steps: first differentiate cot and csc (trig), then apply the derivative of the inner linear function (derivative).qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-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.