Home›Calculus 1›Chain rule›Problem 2.723

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

Problem 2.723 · hard Beautiful Video

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

\[ \frac{d}{d x} \ln{\left(- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \]
derivativeStart with the derivative of the function.✓ 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
\[ = \frac{\frac{d}{d x} \left(- \cot{\left(x + 1 \right)}\right) + \frac{d}{d x} \csc{\left(x + 1 \right)}}{- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
sumApply the sum rule to the inner expression.✓ Proved
\[ = \frac{- \frac{d}{d x} \cot{\left(x + 1 \right)} + \frac{d}{d x} \csc{\left(x + 1 \right)}}{- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
constantFactor out the negative sign.✓ 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 using the chain rule.✓ 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 algebraEvaluate the derivatives of the inner linear terms. Simplify the expression inside the parentheses.✓ Proved
\[ = \csc{\left(x + 1 \right)} \]
algebra simplifyFactor out csc(x + 1) from the numerator and rearrange the denominator. Cancel the common term in the numerator and denominator.✓ Proved

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

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

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 undefined where -cot(x + 1) + csc(x + 1) = 0
3	✓ 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
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
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 (error) — Step 5 applies the derivative to both Ω and σ in one line, violating the one‑rule‑per‑step rule. Step 8 also performs two manipulations (factoring and rearranging) in a single step.
qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives. Each step isolates a single operation, and the labels accurately reflect the rules applied.

Senior review claude-sonnet-5-5, 2026-10-12: pass — The derivation is correct: the derivative of log(csc u - cot u) is (csc² u - csc u cot u)/(csc u - cot u) = csc u. Step 4 is labeled 'constant' where 'constant-multiple' would fit better, which is only a style point.

gpt-oss:20b: dismiss — Step 5 applies a single rule (trig derivative with the chain rule) to the two terms, which counts as one change, and step 8 is a single algebra step: factoring csc out of the numerator and writing the product over the reordered denominator, which equals the original one.
agent:pipeline: uphold — Beautiful nomination: The log of csc(x+1) - cot(x+1), which equals log tan((x+1)/2), differentiates to the simple csc(x+1) after the numerator factors and cancels against the denominator.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — The derivation is correct: the derivative of log(csc u - cot u) is (csc² u - csc u cot u)/(csc u - cot u) = csc u. Step 4 is labeled 'constant' where 'constant-multiple' would fit better, which is only a style point.
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives. Each step isolates a single operation, and the labels accurately reflect the rules applied.
gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies the derivative to both Ω and σ in one line, violating the one‑rule‑per‑step rule. Step 8 also performs two manipulations (factoring and rearranging) in a single step.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies both the chain rule and the trigonometric derivative rule in one line, yet it is labeled only as "trig". Step 4 also incorrectly labels the factoring of a negative sign as "constant"; it should be an "algebra" step. These multi‑rule applications violate the one‑rule‑per‑step requirement.

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.