Derivative of \( \displaystyle \frac{5 \ln{\left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)}}{3} \)
Problem 2.1853 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{5 \ln{\left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)}}{3} \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)}{3 \left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{5 \left(\frac{d}{d x} \left(- \cot{\left(3 x \right)}\right) + \frac{d}{d x} \csc{\left(3 x \right)}\right)}{3 \left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{5 \left(- \frac{d}{d x} \cot{\left(3 x \right)} + \frac{d}{d x} \csc{\left(3 x \right)}\right)}{3 \left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)} \]constant-multipleDistribute the negative sign.✓ Proved
- \[ = \frac{5 \left(- \cot{\left(3 x \right)} \csc{\left(3 x \right)} \frac{d}{d x} 3 x + \csc^{2}{\left(3 x \right)} \frac{d}{d x} 3 x\right)}{3 \left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)} \]trigApply the derivatives for cotangent and cosecant.✓ Proved
- \[ = \frac{5 \left(- 3 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + 3 \csc^{2}{\left(3 x \right)}\right)}{3 \left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)} \]chainApply the chain rule to the inner functions of 3*x.✓ Proved
- \[ = \frac{5 \left(- \cot{\left(3 x \right)} \csc{\left(3 x \right)} + \csc^{2}{\left(3 x \right)}\right)}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]constant-multipleFactor out the 3 from the parentheses and multiply with 5/3.✓ Proved
- \[ = \frac{- 5 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + 5 \csc^{2}{\left(3 x \right)}}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]algebraCombine the terms into a single fraction.✓ Proved
- \[ = 5 \csc{\left(3 x \right)} \]algebra simplifyFactor out csc(3*x) in the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\sin{\left(3 x \right)}} \)
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) + csc(3*x) = 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) + csc(3*x) = 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) + csc(3*x) = 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) + csc(3*x) = 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) + csc(3*x) = 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) + csc(3*x) = 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) + csc(3*x) = 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(3*x) + csc(3*x) = 0 |
| 11 | ✓ 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) = 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 6 applies the derivative of both –cot(3x) and csc(3x) in a single line, combining two separate trig‑derivative applications (and implicitly a chain rule for the inner 3x). According to the contract, each step must change only one thing and use a single rule label; here two rules are applied at once and the label "trig" does not cover the chain rule used for the inner 3x. This violates the step‑granularity rule.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — Step 6 applies the derivative of both –cot(3x) and csc(3x) in a single line, combining two separate trig‑derivative applications (and implicitly a chain rule for the inner 3x). According to the contract, each step must change only one thing and use a single rule label; here two rules are applied at once and the label "trig" does not cover the chain rule used for the inner 3x. This violates the step‑granularity rule.qwen3.6:27b-mlx: pass 2026-10-08gpt-oss:20b: fail (error) 2026-10-08 — Step 6 applies the trigonometric derivative rule twice (once for ‘cot’ and once for ‘csc’) in a single line, violating the one‑rule‑per‑step rule. Similarly, step 7 applies the chain rule twice in one step. These multi‑rule steps make the solution incorrect per the contract.qwen3.6:27b-mlx: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.