Derivative of \( \displaystyle - \frac{\ln{\left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)} \right)}}{4} \)
Problem 2.129 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)} \right)}}{4}\right) \]constant-multipleBring the constant factor out of the derivative.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)} \right)}}{4} \]chainApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)}\right)}{4 \left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)}\right)} \]sumDifferentiate the sum inside the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(4 x - 1 \right)} + \frac{d}{d x} \csc{\left(4 x - 1 \right)}}{4 \left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)}\right)} \]trigDifferentiate the trigonometric functions.✓ Proved
- \[ = - \frac{- 4 \cot{\left(4 x - 1 \right)} \csc{\left(4 x - 1 \right)} - 4 \csc^{2}{\left(4 x - 1 \right)}}{4 \left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)}\right)} \]chainApply the chain rule to the inner functions of cot and csc.✓ Proved
- \[ = - \frac{- \cot{\left(4 x - 1 \right)} \csc{\left(4 x - 1 \right)} - \csc^{2}{\left(4 x - 1 \right)}}{\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)}} \]constant-multiple algebraFactor out the common constant 4. Cancel the 4 in the numerator and denominator.✓ Proved
- \[ = \frac{\cot{\left(4 x - 1 \right)} \csc{\left(4 x - 1 \right)} + \csc^{2}{\left(4 x - 1 \right)}}{\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)}} \]algebraDistribute the negative sign and simplify.✓ Proved
- \[ = \csc{\left(4 x - 1 \right)} \]algebra simplifyFactor out csc(4*x - 1) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\sin{\left(4 x - 1 \right)}} \)
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.
✓ 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.
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(4*x - 1) + csc(4*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(4*x - 1) + csc(4*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(4*x - 1) + csc(4*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(4*x - 1) + csc(4*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(4*x - 1) + csc(4*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(4*x - 1) + csc(4*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(4*x - 1) + csc(4*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(4*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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately describe the operations performed at each stage.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately describe the operations performed at each stage.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 2 incorrectly labels the operation as "chain"; it only applies the constant‑multiple rule, not the chain rule.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 5 applies the chain rule to the inner linear functions, but the label 'chain' is insufficient because the step also relies on the specific derivative formulas for cot and csc (which should be labeled 'trig' or 'derivative'). Step 4 already labeled the differentiation of the trig functions as 'trig', so Step 5 is redundant or mislabeled; it effectively combines the application of the chain rule with the evaluation of the trig derivatives, violating the one-rule-per-step constraint by conflating the structural rule (chain) with the content rule (trig derivatives) which were already separated in Step 4.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 2 incorrectly labels the rule as "chain" and claims the chain rule is applied, yet it only extracts the constant factor; the chain rule is not yet used. Step 3 also mislabels the rule and note, applying both the chain rule and sum rule but only labeling "sum".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.