Derivative of \( \displaystyle - \frac{\ln{\left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)} \right)}}{5} \)
Problem 2.983 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)} \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)} \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)} \right)}}{5} \]constantPull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}\right)}{5 \left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}\right)} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(5 x + 2 \right)} + \frac{d}{d x} \csc{\left(5 x + 2 \right)}}{5 \left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = - \frac{- 5 \cot{\left(5 x + 2 \right)} \csc{\left(5 x + 2 \right)} - 5 \csc^{2}{\left(5 x + 2 \right)}}{5 \left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)}\right)} \]trigApply the derivatives of cot and csc.✓ Proved
- \[ = \csc{\left(5 x + 2 \right)} \]algebra algebra simplifyFactor out common terms in the numerator. Combine the terms. Simplify the expression by canceling common terms.✓ Proved
Answer \( \frac{1}{\sin{\left(5 x + 2 \right)}} \)
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.
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(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 |
| 8 | ✓ 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: pass — The solution correctly applies differentiation rules and simplifies the expression step-by-step. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules and simplifies the expression step-by-step. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the chain rule, sum rule, and trigonometric derivative rules in separate steps. The algebraic simplification is sound and clearly labeled.gpt-oss:20b: fail (error) 2026-09-27 — Step 8 incorrectly simplifies (csc(5*x+2))/(cot(5*x+2)+csc(5*x+2)) to csc(5*x+2); the correct simplification is 1/(cos(5*x+2)+1), not csc. The final answer 1/sin(5*x+2) is also incorrect.
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 2026-09-27 with SymPy 1.14.0.