Derivative of \( \displaystyle \frac{3 \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{3 \ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \)
Problem 2.14 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{3 \ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \).
- \[ \frac{d}{d x} \left(\frac{3 \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{3 \ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10}\right) \]constantStart with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \]constant-multiple algebraFactor out the constant 3/10. Factor out 3/10 from the expression.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{3 \frac{d}{d x} \left(\cos{\left(5 x \right)} + 1\right)}{10 \left(\cos{\left(5 x \right)} + 1\right)} \]chainApply the chain rule to the second term.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{3 \frac{d}{d x} \cos{\left(5 x \right)}}{10 \left(\cos{\left(5 x \right)} + 1\right)} \]sumApply the sum rule to the derivative of the second term.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} + \frac{3 \sin{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} + 1\right)} \]trig algebraDifferentiate cos(5*x). Simplify the signs.✓ Proved
- \[ = \frac{3 \sin{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} + 1\right)} + \frac{3 \frac{d}{d x} \left(\cos{\left(5 x \right)} - 1\right)}{10 \left(\cos{\left(5 x \right)} - 1\right)} \]chainApply the chain rule to the first term.✓ Proved
- \[ = \frac{3 \sin{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} + 1\right)} + \frac{3 \frac{d}{d x} \cos{\left(5 x \right)}}{10 \left(\cos{\left(5 x \right)} - 1\right)} \]sumApply the sum rule to the derivative of the first term.✓ Proved
- \[ = \frac{3 \sin{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} + 1\right)} - \frac{3 \sin{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} - 1\right)} \]trig algebraDifferentiate cos(5*x). Simplify the expression.✓ Proved
- \[ = \frac{3 \left(\frac{1}{\cos{\left(5 x \right)} + 1} - \frac{1}{\cos{\left(5 x \right)} - 1}\right) \sin{\left(5 x \right)}}{2} \]algebra algebraFactor out 5*sin(5*x). Simplify the constant coefficient.✓ Proved
- \[ = - \frac{3 \sin{\left(5 x \right)}}{\left(\cos{\left(5 x \right)} - 1\right) \left(\cos{\left(5 x \right)} + 1\right)} \]algebra algebra algebraFind a common denominator for the terms in the parentheses. Distribute the terms in the numerator. Simplify the numerator.✓ Proved
- \[ = - \frac{3 \sin{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)} - 1} \]algebraExpand the denominator.✓ Proved
- \[ = \frac{3}{\sin{\left(5 x \right)}} \]trig algebra algebra simplifyUse the identity cos(u)**2 - 1 = -sin(u)**2. Simplify the signs. Multiply the terms. Final simplification.✓ Proved
Answer \( \frac{3}{\sin{\left(5 x \right)}} \)
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.
✓ 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(5*x) + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(5*x) + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(5*x) + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(5*x) + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 undefined where cos(5*x)**2 - 1 = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x)**2 - 1 = 0 undefined where sin(5*x) = 0 |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) = 0 |
| 20 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) = 0 |
| 21 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) = 0 |
| 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
deepseek-r1:70b: passgpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-rule constraint, and the labels accurately describe the operations performed.
Every verdict on record (14)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-rule constraint, and the labels accurately describe the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate for the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-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 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18deepseek-r1:70b: pass 2026-09-16
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.