Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.605

Derivative of \( \displaystyle - \frac{\ln{\left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)} \right)}}{2} \)

Problem 2.605 · hard Video

Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)} \right)}}{2} \).

\[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)} \right)}}{2}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)} \right)}}{2} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)}\right)}{2 \left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)}\right)} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = - \frac{\frac{d}{d x} \cot{\left(2 x - 3 \right)} + \frac{d}{d x} \csc{\left(2 x - 3 \right)}}{2 \left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)}\right)} \]
sumApply the sum rule to the inner expression.✓ Proved
\[ = - \frac{- \cot{\left(2 x - 3 \right)} \csc{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right) - \csc^{2}{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 \left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)}\right)} \]
trigDifferentiate the trigonometric functions.✓ Proved
\[ = - \frac{- 2 \cot{\left(2 x - 3 \right)} \csc{\left(2 x - 3 \right)} - 2 \csc^{2}{\left(2 x - 3 \right)}}{2 \left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)}\right)} \]
chain constant-multipleDifferentiate the inner linear function 2x - 3. Factor out the common term 2.✓ Proved
\[ = \frac{\cot{\left(2 x - 3 \right)} \csc{\left(2 x - 3 \right)} + \csc^{2}{\left(2 x - 3 \right)}}{\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)}} \]
algebraSimplify the expression by canceling the -1 and 2.✓ Proved
\[ = \csc{\left(2 x - 3 \right)} \]
algebra simplifyFactor out csc(2*x - 3) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved

Answer \( \frac{1}{\sin{\left(2 x - 3 \right)}} \)

Top: the graph of f = -log(cot(2x - 3) + csc(2x - 3))/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(-2cot(2x - 3)² - 2cot(2x - 3)·csc(2x - 3) - 2)/(2(cot(2x - 3) + csc(2x - 3))).

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(2x - 3) + csc(2x - 3) = 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(2x - 3) + csc(2x - 3) = 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(2x - 3) + csc(2x - 3) = 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(2x - 3) + csc(2x - 3) = 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(2x - 3) + csc(2x - 3) = 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(2x - 3) + csc(2x - 3) = 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(2x - 3) + csc(2x - 3) = 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(2*x - 3) = 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 differentiation rules and algebraic simplifications. Each step isolates a single rule application, and the labels accurately reflect the operations performed.

Senior review claude-sonnet-5-5, 2026-10-10: pass — The panel listed no specific objection, so there is nothing to rule on. The derivation is correct: the inner derivative is -2(csc²+csc·cot), and multiplying by -1/(2(cot+csc)) gives csc(2x-3). Every step has a legitimate rule label and SymPy proved all lines and the final answer.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — The panel listed no specific objection, so there is nothing to rule on. The derivation is correct: the inner derivative is -2(csc²+csc·cot), and multiplying by -1/(2(cot+csc)) gives csc(2x-3). Every step has a legitimate rule label and SymPy proved all lines and the final answer.
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single rule application, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is valid.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed at each stage.
gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies both the chain rule (derivative of the inner function 2*x-3) and the trigonometric derivative rule for cot and csc. It should be split into two separate steps, one for the chain rule and one for the trig derivative.

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.