Derivative of \( \displaystyle \operatorname{atan}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \)
Problem 2.1034 · hard Beautiful Video
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. 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 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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: pass 2026-09-28qwen3.6:27b-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 2026-09-28 with SymPy 1.14.0.