Home›Calculus 1›Chain rule›Problem 2.269

Derivative of \( \displaystyle - \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)} \)

Problem 2.269 · hard Beautiful

Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)} \).

\[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)}\right) \]
derivativeStart 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 negative sign.✓ 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 expression.✓ Proved
\[ = - \frac{- \csc^{2}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) + \frac{d}{d x} \csc{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
trigDifferentiate the cotangent term.✓ Proved
\[ = - \frac{- \cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} - \csc^{2}{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
derivative algebraDifferentiate the cosecant term and the inner linear term. Simplify the expression inside the parentheses.✓ Proved
\[ = \frac{\cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} + \csc^{2}{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
algebraDistribute the negative sign and the fraction.✓ Proved
\[ = \csc{\left(x - 1 \right)} \]
algebra simplifyFactor out csc(x - 1) from the numerator. Cancel the common term in the numerator and denominator.✓ 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.

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 - 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 cot has poles at multiples of pi undefined where cot(x - 1) + csc(x - 1) = 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(x - 1) + csc(x - 1) = 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(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: fail (style) — Step 6 is labeled "derivative" but the operation performed is the trigonometric derivative of σ(x-1) (csc), which already incorporates the derivative of the inner linear term. The correct label should be "trig". This is a labeling defect, not a mathematical error.
deepseek-r1:70b: pass
qwen3.6:27b-mlx: fail (error) — Step 6 applies two differentiation rules simultaneously (derivative of csc and derivative of the inner linear term x-1), violating the one-rule-per-step constraint. Additionally, Step 5 incorrectly labels the application of the chain rule for cot(x-1) as 'trig'; 'trig' is for identities, not differentiation.

Every verdict on record (12)

qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 6 applies two differentiation rules simultaneously (derivative of csc and derivative of the inner linear term x-1), violating the one-rule-per-step constraint. Additionally, Step 5 incorrectly labels the application of the chain rule for cot(x-1) as 'trig'; 'trig' is for identities, not differentiation.
gpt-oss:20b: fail (style) 2026-09-20 — Step 6 is labeled "derivative" but the operation performed is the trigonometric derivative of σ(x-1) (csc), which already incorporates the derivative of the inner linear term. The correct label should be "trig". This is a labeling defect, not a mathematical error.
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 6 applies two differentiation rules simultaneously (derivative of csc and derivative of the inner linear term x-1), violating the one-rule-per-step constraint. Additionally, Step 5 incorrectly labels the differentiation of cot(x-1) as 'trig' when it is a differentiation step requiring 'derivative' (and implicitly 'chain' for the inner function, though the inner derivative is handled in step 6, making step 5's scope ambiguous and its label incorrect for a differentiation action).
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 6 applies two derivative rules in one step: it replaces both Derivative(cot(x-1),x) and Derivative(csc(x-1),x) simultaneously. Each step must change only one thing, so this step violates the contract.
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 6 applies two differentiation rules at once: it differentiates the csc term (trig/chain) and evaluates the derivative of the inner linear term (derivative/constant) in a single step, violating the one-rule-per-step constraint.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: inconclusive 2026-09-17 — reviewer response could not be parsed: {"verdict":"fail","notes":"Step 5 applies both the chain rule (for the inner \(x-1\)) and the derivative of \(\cot\), yet it is labeled only as a “trig” rule. Similarly, step 6 applies the derivative
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.