Derivative of \( \displaystyle - \frac{\ln{\left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)} \right)}}{3} \)
Problem 2.1092 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)} \right)}}{3}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)} \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)}{3 \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cot{\left(3 x - 3 \right)} + \frac{d}{d x} \csc{\left(3 x - 3 \right)}}{3 \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)} \]sumApply the sum rule to the internal terms.✓ Proved
- \[ = - \frac{- \cot{\left(3 x - 3 \right)} \csc{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right) - \csc^{2}{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right)}{3 \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)} \]trigDifferentiate the trigonometric functions.✓ Proved
- \[ = - \frac{- 3 \cot{\left(3 x - 3 \right)} \csc{\left(3 x - 3 \right)} - 3 \csc^{2}{\left(3 x - 3 \right)}}{3 \left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}\right)} \]chain constant-multipleDifferentiate the inner linear function. Factor out the common constant 3.✓ Proved
- \[ = \frac{\cot{\left(3 x - 3 \right)} \csc{\left(3 x - 3 \right)} + \csc^{2}{\left(3 x - 3 \right)}}{\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)}} \]algebra simplifyDistribute the negative sign and rearrange. Cancel the common factor of 3.✓ Proved
- \[ = \csc{\left(3 x - 3 \right)} \]algebra simplifyFactor out csc(3*x - 3) from the numerator. Simplify the fraction.✓ Proved
Answer \( \frac{1}{\sin{\left(3 x - 3 \right)}} \)
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.
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 - 3) + csc(3*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(3*x - 3) + csc(3*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(3*x - 3) + csc(3*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(3*x - 3) + csc(3*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(3*x - 3) + csc(3*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(3*x - 3) + csc(3*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(3*x - 3) + csc(3*x - 3) = 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 - 3) + csc(3*x - 3) = 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 - 3) = 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 algebraic simplifications in single-step increments. The labels used are appropriate for the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels used are appropriate for the operations performed.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-09-28 — Step 9 incorrectly claims to cancel a factor of 3 that no longer exists; the expression already simplifies to (csc(3*x-3)**2 + csc(3*x-3)*cot(3*x-3))/(cot(3*x-3)+csc(3*x-3)) without any 3 to cancel.
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 2026-09-28 with SymPy 1.14.0.