Derivative of \( \displaystyle \ln{\left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)} \right)} \)
Problem 2.1780 · hard Beautiful
Differentiate \( \displaystyle f(x) = \ln{\left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)} \right)} \).
- \[ \frac{d}{d x} \ln{\left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}\right)}{- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- \cot{\left(5 x \right)}\right) + \frac{d}{d x} \csc{\left(5 x \right)}}{- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}} \]sumApply the sum rule to the inner derivative.✓ Proved
- \[ = \frac{- \frac{d}{d x} \cot{\left(5 x \right)} + \frac{d}{d x} \csc{\left(5 x \right)}}{- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}} \]constantFactor out the negative sign.✓ Proved
- \[ = \frac{- 5 \cot{\left(5 x \right)} \csc{\left(5 x \right)} + 5 \csc^{2}{\left(5 x \right)}}{- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}} \]trig algebraDifferentiate the trigonometric functions using the chain rule. Simplify the signs and distribute the constants.✓ Proved
- \[ = 5 \csc{\left(5 x \right)} \]algebra simplifyFactor out 5*csc(5*x) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\sin{\left(5 x \right)}} \)
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 undefined where -cot(5*x) + csc(5*x) = 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(5*x) + csc(5*x) = 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) + csc(5*x) = 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) + csc(5*x) = 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) + csc(5*x) = 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(5*x) + csc(5*x) = 0 |
| 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) = 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)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07gpt-oss:20b: fail (error) 2026-10-07 — Step 5 applies two differentiation rules at once (derivative of cot and derivative of csc, both involving the chain rule). Each step must change only one thing, so this step should be split into two separate derivative applications.qwen3.6:27b-mlx: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.