Derivative of \( \displaystyle - \frac{\ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \)
Problem 2.191 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4}\right) \]constant-multiplePull out the constant factor -1/4.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \]chainApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}\right)}{4 \left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}\right)} \]sumDifferentiate the sum inside the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(4 x - 3 \right)} + \frac{d}{d x} \csc{\left(4 x - 3 \right)}}{4 \left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}\right)} \]chainApply the chain rule to each trigonometric term.✓ Proved
- \[ = - \frac{- 4 \cot{\left(4 x - 3 \right)} \csc{\left(4 x - 3 \right)} - 4 \csc^{2}{\left(4 x - 3 \right)}}{4 \left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}\right)} \]trigDifferentiate cot(u) and csc(u) with respect to x.✓ Proved
- \[ = - \frac{- \cot{\left(4 x - 3 \right)} \csc{\left(4 x - 3 \right)} - \csc^{2}{\left(4 x - 3 \right)}}{\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}} \]algebra algebraFactor out the common constant 4. Cancel the 4 in the numerator with the 1/4 factor.✓ Proved
- \[ = \frac{\cot{\left(4 x - 3 \right)} \csc{\left(4 x - 3 \right)} + \csc^{2}{\left(4 x - 3 \right)}}{\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}} \]algebraDistribute the negative sign and simplify.✓ Proved
- \[ = \csc{\left(4 x - 3 \right)} \]algebra simplifyFactor out csc(4*x - 3) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\sin{\left(4 x - 3 \right)}} \)
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 - 3) + csc(4*x - 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(4*x - 3) + csc(4*x - 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(4*x - 3) + csc(4*x - 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(4*x - 3) + csc(4*x - 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(4*x - 3) + csc(4*x - 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(4*x - 3) + csc(4*x - 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(4*x - 3) + csc(4*x - 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(4*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: fail (style) — Step 3 incorrectly labels the application of the chain rule as a sum; Step 4 incorrectly labels the sum rule as a chain rule. These mislabelings violate the required one‑rule‑per‑step granularity.deepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed, and the final result is correct.
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 reflect the operations performed, and the final result is correct.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule as a sum; Step 4 incorrectly labels the sum rule as a chain rule. These mislabelings violate the required one‑rule‑per‑step granularity.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 4 incorrectly labels the application of the sum rule as "chain"; the step actually uses the sum rule to differentiate the sum inside the logarithm, 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: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 4 incorrectly labels the application of the sum rule as "chain". The step differentiates the sum of two functions, so the correct rule name is "sum", not "chain". No mathematical errors are present.qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 3 labels the rule as "sum" but the step actually applies both the chain rule for the logarithm and the sum rule for differentiating the inner expression. The rule label does not match the operations performed.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.