Derivative of \( \displaystyle - \frac{5 \ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \)
Problem 2.48 · hard
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4}\right) \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}\right)}{4 \left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}\right)} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = - \frac{5 \left(\frac{d}{d x} \cot{\left(4 x - 3 \right)} + \frac{d}{d x} \csc{\left(4 x - 3 \right)}\right)}{4 \left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}\right)} \]sumApply the sum rule to the inner terms.✓ Proved
- \[ = - \frac{5 \left(- \cot{\left(4 x - 3 \right)} \csc{\left(4 x - 3 \right)} \frac{d}{d x} \left(4 x - 3\right) - \csc^{2}{\left(4 x - 3 \right)} \frac{d}{d x} \left(4 x - 3\right)\right)}{4 \left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}\right)} \]trigDifferentiate the trigonometric functions.✓ Proved
- \[ = - \frac{5 \left(- 4 \cot{\left(4 x - 3 \right)} \csc{\left(4 x - 3 \right)} - 4 \csc^{2}{\left(4 x - 3 \right)}\right)}{4 \left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}\right)} \]chainApply the chain rule to the inner linear function.✓ Proved
- \[ = - \frac{5 \left(- \cot{\left(4 x - 3 \right)} \csc{\left(4 x - 3 \right)} - \csc^{2}{\left(4 x - 3 \right)}\right)}{\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}} \]algebra algebraFactor out the common constant 4. Multiply the constants -5/4 and 4.✓ Proved
- \[ = \frac{5 \cot{\left(4 x - 3 \right)} \csc{\left(4 x - 3 \right)} + 5 \csc^{2}{\left(4 x - 3 \right)}}{\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)}} \]algebra algebraCombine the expression into a single fraction. Distribute the negative sign in the numerator.✓ Proved
- \[ = 5 \csc{\left(4 x - 3 \right)} \]algebra simplifyFactor out csc(4*x - 3) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\sin{\left(4 x - 3 \right)}} \)
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.
✓ 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 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(4*x - 3) + csc(4*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(4*x - 3) + csc(4*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(4*x - 3) + csc(4*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(4*x - 3) + csc(4*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(4*x - 3) + csc(4*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(4*x - 3) + csc(4*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(4*x - 3) + csc(4*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(4*x - 3) + csc(4*x - 3) = 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 cot(4*x - 3) + csc(4*x - 3) = 0 |
| 12 | ✓ 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(4*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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed, and the final simplification is valid.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed, and the final simplification is valid.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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 2026-09-26 with SymPy 1.14.0.