Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.454

Derivative of \( \displaystyle - 2 \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \)

Problem 2.454 · hard Beautiful

Differentiate \( \displaystyle f(x) = - 2 \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \).

\[ \frac{d}{d x} \left(- 2 \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - 2 \frac{d}{d x} \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{2 \frac{d}{d x} \left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}\right)}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
chainApply the chain rule for the natural logarithm.✓ Proved
\[ = - \frac{2 \left(\frac{d}{d x} \cot{\left(x + 1 \right)} + \frac{d}{d x} \csc{\left(x + 1 \right)}\right)}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
sumApply the sum rule to the inner expression.✓ Proved
\[ = - \frac{2 \left(- \csc^{2}{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) + \frac{d}{d x} \csc{\left(x + 1 \right)}\right)}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
trigDifferentiate the cotangent function.✓ Proved
\[ = - \frac{2 \left(- \csc^{2}{\left(x + 1 \right)} + \frac{d}{d x} \csc{\left(x + 1 \right)}\right)}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
chainDifferentiate the inner linear term x + 1.✓ Proved
\[ = - \frac{2 \left(- \cot{\left(x + 1 \right)} \csc{\left(x + 1 \right)} - \csc^{2}{\left(x + 1 \right)}\right)}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
trig chainDifferentiate the cosecant function. Evaluate the derivative of the inner term.✓ Proved
\[ = \frac{2 \cot{\left(x + 1 \right)} \csc{\left(x + 1 \right)} + 2 \csc^{2}{\left(x + 1 \right)}}{\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
algebra algebraCombine the terms into a single fraction. Distribute the negative sign and the constant 2.✓ Proved
\[ = 2 \csc{\left(x + 1 \right)} \]
algebra simplifyFactor out csc(x + 1) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved

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

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 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 cot has poles at multiples of pi undefined where cot(x + 1) + csc(x + 1) = 0
11	✓ 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
12	✓ 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 8 is labeled "chain" but only performs an algebraic simplification; the chain rule was not applied there. The label should be "algebra" or omitted.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: fail (style) 2026-09-21 — Step 8 is labeled "chain" but only performs an algebraic simplification; the chain rule was not applied there. The label should be "algebra" or omitted.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (style) 2026-09-20 — Step 8 is labeled "chain" but no chain rule is applied there; the expression is already a single fraction. This mislabeling violates the rule‑granularity requirement.
qwen3.6:27b-mlx: pass 2026-09-20
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.