Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{2} - \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{2} \)
Problem 2.1302 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{2} - \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(\frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{2} - \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{2}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{2} - \frac{d}{d x} \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{2} \]sumApply the difference rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{2} - \frac{\frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{2} \]constant-multipleFactor out the constant 1/2.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cos{\left(5 x \right)} + 1\right)}{2 \left(\cos{\left(5 x \right)} + 1\right)} + \frac{\frac{d}{d x} \left(\cos{\left(5 x \right)} - 1\right)}{2 \left(\cos{\left(5 x \right)} - 1\right)} \]chainApply the chain rule to each logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} + 1\right)} + \frac{\frac{d}{d x} \cos{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} - 1\right)} \]derivativeThe derivative of the constant -1 and +1 is zero.✓ Proved
- \[ = \frac{5 \sin{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} + 1\right)} - \frac{5 \sin{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} - 1\right)} \]trig algebraDifferentiate the cosine function. Distribute the constants and simplify signs.✓ Proved
- \[ = \frac{5 \left(\frac{1}{\cos{\left(5 x \right)} + 1} - \frac{1}{\cos{\left(5 x \right)} - 1}\right) \sin{\left(5 x \right)}}{2} \]algebraFactor out the common term 5/2 * sin(5*x).✓ Proved
- \[ = - \frac{5 \sin{\left(5 x \right)}}{\left(\cos{\left(5 x \right)} - 1\right) \left(\cos{\left(5 x \right)} + 1\right)} \]algebraCombine the fractions using a common denominator.✓ Proved
- \[ = - \frac{5 \sin{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)} - 1} \]algebra algebraSimplify the numerator and expand the denominator. Simplify the constant coefficients.✓ Proved
- \[ = \frac{5}{\sin{\left(5 x \right)}} \]trig simplifyUse the identity cos(x)**2 - 1 = -sin(x)**2. Cancel the common sin(5*x) term and simplify signs.✓ Proved
- \[ = 5 \csc{\left(5 x \right)} \]simplifyRewrite using the cosecant function.✓ 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.
✓ 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 undefined where cos(5*x) + 1 = 0 |
| 5 | ✓ 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 |
| 6 | ✓ 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 |
| 7 | ✓ 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 |
| 8 | ✓ 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 |
| 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 undefined where cos(5*x)**2 - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x)**2 - 1 = 0 |
| 12 | ✓ 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 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) = 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 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.