Derivative of \( \displaystyle \frac{\ln{\left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)} \right)}}{5} \)
Problem 2.991 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)} \right)}}{5} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)} \right)}}{5} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)} \right)}}{5} \]constant-multiplePull out the constant factor 1/5.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \cot{\left(5 x - 3 \right)} + \frac{d}{d x} \csc{\left(5 x - 3 \right)}}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{- \cot{\left(5 x - 3 \right)} \csc{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x - 3\right) - \csc^{2}{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x - 3\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]trigApply the derivatives of cot and csc.✓ Proved
- \[ = \frac{- 5 \cot{\left(5 x - 3 \right)} \csc{\left(5 x - 3 \right)} - 5 \csc^{2}{\left(5 x - 3 \right)}}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]chainDifferentiate the inner linear function 5x - 3.✓ Proved
- \[ = \frac{- \cot{\left(5 x - 3 \right)} \csc{\left(5 x - 3 \right)} - \csc^{2}{\left(5 x - 3 \right)}}{\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}} \]constant-multiple simplifyFactor out the constant 5. Cancel the 5 and the 1/5 and simplify the expression.✓ Proved
- \[ = - \csc{\left(5 x - 3 \right)} \]simplifySimplify the trigonometric fraction using identities.✓ Proved
Answer \( - \frac{1}{\sin{\left(5 x - 3 \right)}} \)
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(5*x - 3) + csc(5*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(5*x - 3) + csc(5*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(5*x - 3) + csc(5*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(5*x - 3) + csc(5*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(5*x - 3) + csc(5*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(5*x - 3) + csc(5*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(5*x - 3) + csc(5*x - 3) = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(5*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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (error) 2026-09-27 — Step 5 applies two separate trigonometric derivative rules (for cot and for csc) in a single line, violating the rule that each step must change only one thing. This combines two operations into one step.
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.