Derivative of \( \displaystyle - \ln{\left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)} \right)} \)
Problem 2.579 · hard Video
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)} \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)} \right)} \]constantPull out the constant factor -1.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}\right)}{\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(5 x + 2 \right)} + \frac{d}{d x} \csc{\left(5 x + 2 \right)}}{\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}} \]sumApply the sum rule to the inner expression.✓ Proved
- \[ = - \frac{- \cot{\left(5 x + 2 \right)} \csc{\left(5 x + 2 \right)} \frac{d}{d x} \left(5 x + 2\right) - \csc^{2}{\left(5 x + 2 \right)} \frac{d}{d x} \left(5 x + 2\right)}{\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}} \]trigDifferentiate the trigonometric functions using the chain rule.✓ Proved
- \[ = - \frac{- 5 \cot{\left(5 x + 2 \right)} \csc{\left(5 x + 2 \right)} - 5 \csc^{2}{\left(5 x + 2 \right)}}{\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}} \]derivative algebraDifferentiate the linear term 5*x + 2. Factor out the common term -5.✓ Proved
- \[ = \frac{5 \cot{\left(5 x + 2 \right)} \csc{\left(5 x + 2 \right)} + 5 \csc^{2}{\left(5 x + 2 \right)}}{\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}} \]algebraSimplify the signs and the fraction.✓ Proved
- \[ = 5 \csc{\left(5 x + 2 \right)} \]algebra simplifyFactor out csc(5*x + 2) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\sin{\left(5 x + 2 \right)}} \)
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.
✓ 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. How we verify
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(5*x + 2) + csc(5*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(5*x + 2) + csc(5*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(5*x + 2) + csc(5*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(5*x + 2) + csc(5*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(5*x + 2) + csc(5*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(5*x + 2) + csc(5*x + 2) = 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(5*x + 2) + csc(5*x + 2) = 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(5*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: passqwen3.6:27b-mlx: fail (style) — Step 5 applies the chain rule to differentiate the inner trigonometric functions but is labeled 'trig'. The label 'chain' is available in the vocabulary and more accurately describes the application of the chain rule here. Additionally, Step 3 is labeled 'logarithmic' but the note says 'Apply the chain rule'; while 'logarithmic' is acceptable for the derivative of log, the note is slightly imprecise, though less severe than the labeling error in Step 5.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 5 applies the chain rule to differentiate the inner trigonometric functions but is labeled 'trig'. The label 'chain' is available in the vocabulary and more accurately describes the application of the chain rule here. Additionally, Step 3 is labeled 'logarithmic' but the note says 'Apply the chain rule'; while 'logarithmic' is acceptable for the derivative of log, the note is slightly imprecise, though less severe than the labeling error in Step 5.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-21 — 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: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives in separate steps. The algebraic simplifications are valid and clearly labeled.gpt-oss:20b: pass 2026-09-21
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.