Derivative of \( \displaystyle \operatorname{atan}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \)
Problem 2.1034 · hard Beautiful
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \).
- \[ \frac{d}{d x} \operatorname{atan}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]chainApply the chain rule for the arctangent function.✓ Proved
- \[ = \frac{\frac{d}{d x} \cot{\left(3 x \right)} + \frac{d}{d x} \csc{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]sumApply the sum rule to the inner derivative.✓ Proved
- \[ = \frac{- 3 \csc^{2}{\left(3 x \right)} + \frac{d}{d x} \csc{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]derivativeDifferentiate the cotangent term.✓ Proved
- \[ = \frac{- 3 \cot{\left(3 x \right)} \csc{\left(3 x \right)} - 3 \csc^{2}{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]derivativeDifferentiate the cosecant term.✓ Proved
- \[ = - \frac{3 \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right) \csc{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]algebra algebraFactor out common terms in the numerator. Combine the expression into a single fraction.✓ Proved
- \[ = - \frac{3 \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right) \csc{\left(3 x \right)}}{\cot^{2}{\left(3 x \right)} + 2 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + \csc^{2}{\left(3 x \right)} + 1} \]algebraExpand the squared term in the denominator.✓ Proved
- \[ = - \frac{3 \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right) \csc{\left(3 x \right)}}{2 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + 2 \csc^{2}{\left(3 x \right)}} \]algebra simplifyUse the identity cot(3*x)**2 = csc(3*x)**2 - 1. Simplify the denominator by combining like terms.✓ Proved
- \[ = - \frac{3}{2} \]algebra simplify simplifyFactor the denominator. Cancel the common factor (csc(3*x) + cot(3*x)). Cancel the common factor csc(3*x) to get the final result.✓ Proved
Answer \( - \frac{3}{2} \)
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 csc has poles at multiples of pi cot has poles at multiples of pi undefined where (cot(3*x) + csc(3*x))**2 + 1 = 0 |
| 3 | ✓ 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) + csc(3*x))**2 + 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) + csc(3*x))**2 + 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) + csc(3*x))**2 + 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) + csc(3*x))**2 + 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) + csc(3*x))**2 + 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) + csc(3*x))**2 + 1 = 0 undefined where cot(3*x)**2 + 2*cot(3*x)*csc(3*x) + csc(3*x)**2 + 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)**2 + 2*cot(3*x)*csc(3*x) + csc(3*x)**2 + 1 = 0 undefined where 2*cot(3*x)*csc(3*x) + 2*csc(3*x)**2 = 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 2*cot(3*x)*csc(3*x) + 2*csc(3*x)**2 = 0 |
| 11 | ✓ 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 2*cot(3*x)*csc(3*x) + 2*csc(3*x)**2 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 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 the chain rule, sum rule, and standard trigonometric derivatives. The algebraic simplification steps are valid, and the labels used are consistent with the provided vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the chain rule, sum rule, and standard trigonometric derivatives. The algebraic simplification steps are valid, and the labels used are consistent with the provided vocabulary.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27
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.