Derivative of \( \displaystyle \frac{\ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \)
Problem 2.479 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \]constant-multiplePull out the constant factor 1/3.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \]chainApply the chain rule to the natural logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{\frac{d}{d x} \cot{\left(3 x - 1 \right)} + \frac{d}{d x} \csc{\left(3 x - 1 \right)}}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]trigApply the derivative formulas for cotangent and cosecant.✓ Proved
- \[ = \frac{- \cot{\left(3 x - 1 \right)} \csc{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right) - \csc^{2}{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right)}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]derivativeEvaluate the derivatives of the inner linear functions.✓ Proved
- \[ = \frac{- 3 \cot{\left(3 x - 1 \right)} \csc{\left(3 x - 1 \right)} - 3 \csc^{2}{\left(3 x - 1 \right)}}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]constant-multipleFactor out the common 3 from the derivative terms.✓ Proved
- \[ = \frac{- \cot{\left(3 x - 1 \right)} \csc{\left(3 x - 1 \right)} - \csc^{2}{\left(3 x - 1 \right)}}{\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}} \]simplify algebraCancel the 3 and 1/3. Factor out -csc(3*x - 1) from the numerator.✓ Proved
- \[ = - \csc{\left(3 x - 1 \right)} \]simplify simplifyCancel the common term in the numerator and denominator. Final simplified result.✓ Proved
Answer \( - \frac{1}{\sin{\left(3 x - 1 \right)}} \)
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(3*x - 1) + csc(3*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(3*x - 1) + csc(3*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(3*x - 1) + csc(3*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(3*x - 1) + csc(3*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(3*x - 1) + csc(3*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(3*x - 1) + csc(3*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(3*x - 1) + csc(3*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(3*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: passqwen3.6:27b-mlx: fail (error) — Step 5 is labeled 'derivative' but performs the chain rule expansion for cot and csc (introducing Derivative(3*x - 1, x)). The label 'derivative' is reserved for unfolding d/dx on known forms (like d/dx(sin x) = cos x), not for applying the chain rule to composite functions. Additionally, Step 3 is labeled 'sum' but actually applies the chain rule to the logarithm; the sum rule is applied in Step 4.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 5 is labeled 'derivative' but performs the chain rule expansion for cot and csc (introducing Derivative(3*x - 1, x)). The label 'derivative' is reserved for unfolding d/dx on known forms (like d/dx(sin x) = cos x), not for applying the chain rule to composite functions. Additionally, Step 3 is labeled 'sum' but actually applies the chain rule to the logarithm; the sum rule is applied in Step 4.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 5 applies the chain rule to differentiate the inner trigonometric functions, but is labeled 'derivative', which is reserved for basic derivatives (like d/dx of x or sin x). Step 8 performs algebraic factoring but is labeled 'algebra' while the note describes factoring; however, the primary defect is Step 5 mislabeling the chain rule application as a basic derivative.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the chain rule as "sum"; it should be "chain". Step 4 incorrectly labels the sum rule as "trig"; it should be "sum". These labeling errors violate the required granularity of the solution steps.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and simplifies the expression step-by-step. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (error) 2026-09-20 — Step 6 applies two rules at once (derivative of the linear inner function and factoring out the common factor 3). It also labels the operation as "constant-multiple" even though the derivative of 3*x-1 is a separate rule. This violates 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.