Derivative of \( \displaystyle - \frac{\ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3} \)
Problem 2.681 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3} \]constantPull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(3 x + 1 \right)} + \frac{d}{d x} \csc{\left(3 x + 1 \right)}}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = - \frac{- \cot{\left(3 x + 1 \right)} \csc{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right) - \csc^{2}{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right)}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]trigApply the derivatives for cot and csc.✓ Proved
- \[ = - \frac{- 3 \cot{\left(3 x + 1 \right)} \csc{\left(3 x + 1 \right)} - 3 \csc^{2}{\left(3 x + 1 \right)}}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]chainDifferentiate the inner linear function 3x + 1.✓ Proved
- \[ = - \frac{- \cot{\left(3 x + 1 \right)} \csc{\left(3 x + 1 \right)} - \csc^{2}{\left(3 x + 1 \right)}}{\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}} \]constant-multiple algebraFactor out the constant 3. Cancel the 3 and the 1/3.✓ Proved
- \[ = \frac{\cot{\left(3 x + 1 \right)} \csc{\left(3 x + 1 \right)} + \csc^{2}{\left(3 x + 1 \right)}}{\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}} \]algebraDistribute the negative sign.✓ Proved
- \[ = \csc{\left(3 x + 1 \right)} \]algebra simplifyFactor out csc(3*x + 1) from the numerator. Simplify the fraction.✓ Proved
Answer \( \frac{1}{\sin{\left(3 x + 1 \right)}} \)
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.
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 + 1) + csc(3*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(3*x + 1) + csc(3*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(3*x + 1) + csc(3*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(3*x + 1) + csc(3*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(3*x + 1) + csc(3*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(3*x + 1) + csc(3*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(3*x + 1) + csc(3*x + 1) = 0 |
| 10 | ✓ 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 + 1) + csc(3*x + 1) = 0 |
| 11 | ✓ 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(3*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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules and algebraic simplifications in distinct steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies the trig derivative to both cot and csc terms at once, violating the rule that each step must change only one thing. The correct approach would split this into two separate steps, each applying the trig rule to a single term.
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.