Home›Calculus 1›Chain rule›Problem 2.247

Derivative of \( \displaystyle - \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)} \)

Problem 2.247 · hard Beautiful Video

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 \( \frac{1}{\sin{\left(x + 2 \right)}} \)

Top: the graph of f = -log(cot(x + 2) + csc(x + 2)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(-cot(x + 2)² - cot(x + 2)·csc(x + 2) - 1)/(cot(x + 2) + csc(x + 2)).

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. 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 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: pass
deepseek-r1:70b: pass
qwen3.6:27b-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.

Senior review claude-sonnet-5-5, 2026-10-06: pass — Every line was proved equivalent by SymPy, and the result csc(x+2) equals the stated answer 1/sin(x+2). The rule labels are legitimate. The line 8 note says 'cancel' although that line only merges the two negatives into one fraction and the cancellation happens in line 9, which is a harmless wording slip.

agent:pipeline: uphold — Beautiful nomination: The derivative of -log(cot u + csc u) looks messy, but the numerator is csc u times the denominator, so everything cancels to csc u.

Every verdict on record (13)

claude-sonnet-5-5: pass 2026-10-06 — Every line was proved equivalent by SymPy, and the result csc(x+2) equals the stated answer 1/sin(x+2). The rule labels are legitimate. The line 8 note says 'cancel' although that line only merges the two negatives into one fraction and the cancellation happens in line 9, which is a harmless wording slip.
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 2026-09-20
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 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 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-17
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.