Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.1191

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

Problem 2.1191 · hard Video

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

\[ \frac{d}{d x} \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)} \]
derivativeStart with the derivative of the function.✓ 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 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 terms.✓ Proved
\[ = \frac{- \csc^{2}{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right) + \frac{d}{d x} \csc{\left(x + 2 \right)}}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]
trigDifferentiate the cotangent term.✓ Proved
\[ = \frac{- \csc^{2}{\left(x + 2 \right)} + \frac{d}{d x} \csc{\left(x + 2 \right)}}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]
chain algebraDifferentiate the inner part of the cotangent derivative. Simplify the derivative of the constant 2.✓ Proved
\[ = \frac{- \cot{\left(x + 2 \right)} \csc{\left(x + 2 \right)} - \csc^{2}{\left(x + 2 \right)}}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]
trigDifferentiate the cosecant term.✓ Proved
\[ = - \csc{\left(x + 2 \right)} \]
algebra simplifyFactor out csc(x + 2) from the numerator. Cancel the common term in the numerator and denominator.✓ 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 undefined where cot(x + 2) + csc(x + 2) = 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 + 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 cot has poles at multiples of pi undefined where cot(x + 2) + csc(x + 2) = 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 + 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
qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives in separate steps. The final simplification is algebraically sound and the labels are appropriate.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives in separate steps. The final simplification is algebraically sound and the labels are appropriate.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives in a step-by-step manner. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-09-29

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 2026-10-03 with SymPy 1.14.0.