Derivative of \( \displaystyle - \frac{\ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} + \frac{\ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \)
Problem 2.1205 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} + \frac{\ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} + \frac{\ln{\left(\cos^{2}{\left(x \right)} \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} + \frac{\frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \]sumSplit the derivative into two parts.✓ Proved
- \[ = \frac{\frac{d}{d x} \cos^{2}{\left(x \right)}}{4 \cos^{2}{\left(x \right)}} - \frac{\frac{d}{d x} \left(\cos^{2}{\left(x \right)} - 1\right)}{4 \left(\cos^{2}{\left(x \right)} - 1\right)} \]chainApply the chain rule to both logarithmic terms.✓ Proved
- \[ = \frac{\frac{d}{d x} \cos{\left(x \right)}}{2 \cos{\left(x \right)}} - \frac{\cos{\left(x \right)} \frac{d}{d x} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right)} \]chainApply the chain rule to the squared cosine terms.✓ Proved
- \[ = - \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right)} \]trig algebraDifferentiate cos(x). Simplify the products in the numerators.✓ Proved
- \[ = - \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{4 \cos^{2}{\left(x \right)} - 4} \]algebraDistribute the negative sign and simplify constants.✓ Proved
- \[ = - \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} - \frac{\cos{\left(x \right)}}{2 \sin{\left(x \right)}} \]algebra algebra simplify algebraUse the identity cos(x)**2 - 1 = -sin(x)**2. Simplify the signs. Cancel common factors of sin(x) and cos(x). Factor out -1/2.✓ Proved
- \[ = - \frac{\tan{\left(x \right)}}{2} - \frac{1}{2 \tan{\left(x \right)}} \]rewriteRewrite cot(x) as 1/tan(x).✓ Proved
- \[ = - \frac{\tan{\left(x \right)}}{2} - \frac{\cot{\left(x \right)}}{2} \]simplifySimplify the expression.✓ Proved
- \[ = - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \sin{\left(x \right)} \cos{\left(x \right)}} \]algebraCombine the fractions using a common denominator.✓ Proved
- \[ = - \frac{1}{2 \sin{\left(x \right)} \cos{\left(x \right)}} \]simplify algebraUse the identity cos(x)**2 + sin(x)**2 = 1. Distribute the 1/2.✓ Proved
- \[ = - \frac{1}{\sin{\left(2 x \right)}} \]rewriteUse the double angle identity sin(2x) = 2sin(x)cos(x).✓ Proved
Answer \( - \frac{1}{\sin{\left(2 x \right)}} \)
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 undefined where cos(x)**2 - 1 = 0 undefined where cos(x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x)**2 - 1 = 0 undefined where cos(x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x)**2 - 1 = 0 undefined where cos(x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x)**2 - 1 = 0 undefined where cos(x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x)**2 - 1 = 0 undefined where cos(x) = 0 undefined where 4*cos(x)**2 - 4 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*cos(x)**2 - 4 = 0 undefined where cos(x) = 0 undefined where sin(x) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 0 undefined where cos(x) = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 0 undefined where cos(x) = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 0 undefined where cos(x) = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 0 undefined where cos(x) = 0 tan has poles at odd multiples of pi/2 undefined where tan(x) = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x) = 0 cot has poles at multiples of pi |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 cot has poles at multiples of pi undefined where sin(x) = 0 undefined where cos(x) = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 0 undefined where cos(x) = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 0 undefined where cos(x) = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 0 undefined where cos(x) = 0 undefined where sin(2*x) = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(2*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 2026-09-29 — 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 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed, and the final result matches the stated answer.gpt-oss:20b: fail (style) 2026-09-29 — Each step should modify only one term or apply a single rule. Steps 3 and 4 apply the chain rule to two terms at once, violating the granularity requirement.
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 2026-09-29 with SymPy 1.14.0.