Derivative of \( \displaystyle - \frac{\ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)}}{3} \)
Problem 2.99 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)}}{3}\right) \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)}}{3} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}\right)}{3 \left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}\right)} \]sumApply the sum rule to the derivative.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(3 x + 2 \right)} + \frac{d}{d x} \csc{\left(3 x + 2 \right)}}{3 \left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}\right)} \]chainApply the chain rule to each trigonometric term.✓ Proved
- \[ = - \frac{- 3 \cot{\left(3 x + 2 \right)} \csc{\left(3 x + 2 \right)} - 3 \csc^{2}{\left(3 x + 2 \right)}}{3 \left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}\right)} \]derivative algebraDifferentiate the inner functions. Factor out the common term 3.✓ Proved
- \[ = \frac{\cot{\left(3 x + 2 \right)} \csc{\left(3 x + 2 \right)} + \csc^{2}{\left(3 x + 2 \right)}}{\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}} \]simplifyCancel the -1/3 and -3 terms.✓ Proved
- \[ = \csc{\left(3 x + 2 \right)} \]simplifyFactor and simplify the trigonometric expression.✓ Proved
Answer \( \frac{1}{\sin{\left(3 x + 2 \right)}} \)
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 + 2) + csc(3*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(3*x + 2) + csc(3*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(3*x + 2) + csc(3*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(3*x + 2) + csc(3*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(3*x + 2) + csc(3*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(3*x + 2) + csc(3*x + 2) = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(3*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: fail (style) — Step 3 incorrectly labels the application of the chain rule for the logarithm as "sum", and step 4 incorrectly labels the application of the sum rule for the inner derivative as "chain". These mislabelings violate the rule‑granularity requirement.deepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule for the logarithm as "sum", and step 4 incorrectly labels the application of the sum rule for the inner derivative as "chain". These mislabelings violate the rule‑granularity requirement.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 applies the chain rule to the logarithm, but the label 'chain' is insufficiently specific given the vocabulary includes 'logarithmic'; more critically, Step 4 applies the chain rule to two separate terms simultaneously, violating the 'one rule per step' constraint (it should be split into two steps or labeled 'sum' if treating the derivative of the sum, but the note claims chain rule on 'each' term, implying two operations). Step 5 labels the differentiation of cot and csc as 'derivative', which is correct, but Step 4's label 'chain' is applied to a step that actually performs the differentiation of the inner functions' derivatives (which happens in step 5) or is just a placeholder. Actually, looking closely: Step 4 expands the derivative of the sum into sum of derivatives? No, Step 3 already did that. Step 4 keeps the derivative operator on the terms. Step 5 removes the derivative operator. So Step 4 is effectively a no-op or a re-grouping, but labeled 'chain'. The real issue is Step 4: it claims to apply chain rule, but the chain rule application (differentiating the outer function) happened in Step 2/3. Step 4 just sits there. Wait, Step 3 has `Derivative(cot(...) + csc(...), x)`. Step 4 has `Derivative(cot(...), x) + Derivative(csc(...), x)`. This is the linearity/sum rule, but Step 3 was labeled 'sum'. Step 4 is labeled 'chain'. This is a mislabeling. Step 4 does not apply the chain rule; it just separates the terms (which was arguably done in 3, or 3 was the sum rule and 4 is redundant/wrongly labeled). Let's look at 3->4. 3: `... * Derivative(cot + csc, x)`. 4: `... * (Derivative(cot, x) + Derivative(csc, x))`. This is applying the sum rule to the derivative. Step 3 was labeled 'sum' but the expression in 3 still has `Derivative(cot+csc, x)`. So Step 3 didn't apply the sum rule to the derivative yet? Step 3 note says 'Apply the sum rule to the derivative'. But the expression in 3 is `... * Derivative(cot+csc, x)`. The expression in 4 is `... * (Derivative(cot, x) + Derivative(csc, x))`. So the transition 3->4 is the sum rule. But Step 3 is labeled 'sum' and Step 4 is labeled 'chain'. This is a mess. Step 3 label 'sum' is wrong because the expression didn't change to a sum of derivatives. Step 4 label 'chain' is wrong because it's a sum rule application. Also, Step 2 label 'chain' is acceptable for log, but 'logarithmic' is in the vocab. The prompt says 'A label is correct if it names a rule the step legitimately applied'. Step 2 applies chain rule (derivative of log is 1/u * u'). So 'chain' is okay. Step 3: Expression is `-1/3 * (1/(...)) * Derivative(cot+csc, x)`. This is the result of applying chain rule to log. So Step 2->3 is the chain rule application. Step 2 label 'chain' is correct. Step 3 label 'sum' is incorrect because the expression in 3 is not a sum of derivatives yet. Step 4 label 'chain' is incorrect because the transition 3->4 is the sum rule. So we have mislabeled steps.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule as "sum"; the correct label should be "chain". Likewise, step 4 applies the sum rule to the derivative of the inner sum, so its label should be "sum" instead of "chain".qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 is labeled 'derivative' but applies the chain rule to differentiate the composite trigonometric functions (e.g., d/dx cot(3x+2)), which requires the 'chain' rule. The 'derivative' rule is reserved for basic derivatives like d/dx sin(u) = cos(u) * u', but here the inner derivative of (3x+2) is also computed, making it a chain rule application. Furthermore, Step 4 is labeled 'chain' but merely sets up the differentiation of the sum, while Step 5 does the actual differentiation; the labeling is inconsistent and incorrect regarding the specific rules applied at each stage.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Steps 2, 3, and 4 have incorrect rule labels: step 2 is just a constant‑multiple extraction, not a chain rule; step 3 applies the chain rule to the log, not the sum rule; step 4 applies the sum rule to the derivative of the inner sum, not the chain rule.deepseek-r1:70b: pass 2026-09-17
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.