Derivative of \( \displaystyle \frac{3 \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{3 \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \)
Problem 2.528 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{3 \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{3 \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{3 \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4}\right) \]algebraDifferentiate the function with respect to x. Factor out the common constant 3/4.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\ln{\left(\cos^{2}{\left(x \right)} - 1 \right)} - \ln{\left(\cos^{2}{\left(x \right)} \right)}\right)}{4} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{3 \frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \]sumApply the difference rule.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{3 \frac{d}{d x} \cos^{2}{\left(x \right)}}{4 \cos^{2}{\left(x \right)}} \]chainApply the chain rule to the second term.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{3 \frac{d}{d x} \cos{\left(x \right)}}{2 \cos{\left(x \right)}} \]chain algebraApply the chain rule to the inner function cos(x)**2. Simplify the expression inside the parentheses.✓ Proved
- \[ = \frac{3 \sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{3 \frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} \]trigDifferentiate cos(x).✓ Proved
- \[ = \frac{3 \tan{\left(x \right)}}{2} + \frac{3 \frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} \]algebraSimplify the expression using tan(x) = sin(x)/cos(x).✓ Proved
- \[ = \frac{3 \tan{\left(x \right)}}{2} + \frac{3 \frac{d}{d x} \left(\cos^{2}{\left(x \right)} - 1\right)}{4 \left(\cos^{2}{\left(x \right)} - 1\right)} \]chainApply the chain rule to the first term.✓ Proved
- \[ = \frac{3 \tan{\left(x \right)}}{2} + \frac{3 \cos{\left(x \right)} \frac{d}{d x} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right)} \]chainApply the chain rule to the inner function cos(x)**2 - 1.✓ Proved
- \[ = \frac{3 \tan{\left(x \right)}}{2} - \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right)} \]trig algebraDifferentiate cos(x). Simplify the expression.✓ Proved
- \[ = \frac{3 \tan{\left(x \right)}}{2} - \frac{3 \sin{\left(2 x \right)}}{4 \left(\cos^{2}{\left(x \right)} - 1\right)} \]trig algebraUse the double angle identity sin(2x) = 2sin(x)cos(x). Simplify the fraction.✓ Proved
- \[ = \frac{3 \tan{\left(x \right)}}{2} + \frac{3 \sin{\left(2 x \right)}}{4 \sin^{2}{\left(x \right)}} \]algebraUse the identity cos(x)**2 - 1 = -sin(x)**2.✓ Proved
- \[ = \frac{3 \tan{\left(x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2 \sin{\left(x \right)}} \]algebra algebraSimplify the fraction by canceling the negative signs. Cancel one sin(x) from the numerator and denominator.✓ Proved
- \[ = \frac{3 \tan{\left(x \right)}}{2} + \frac{3 \cot{\left(x \right)}}{2} \]trig algebraUse the identity cot(x) = cos(x)/sin(x). Factor out the 2 and multiply by 3/4.✓ Proved
- \[ = \frac{3 \tan{\left(x \right)}}{2} + \frac{3}{2 \tan{\left(x \right)}} \]trigUse the identity cot(x) = 1/tan(x).✓ Proved
- \[ = \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{2 \tan{\left(x \right)}} \]algebraFind a common denominator.✓ Proved
- \[ = \frac{3 \sec^{2}{\left(x \right)}}{2 \tan{\left(x \right)}} \]trigUse the identity 1 + tan(x)**2 = sec(x)**2.✓ Proved
- \[ = \frac{3 \cos{\left(x \right)} \sec^{2}{\left(x \right)}}{2 \sin{\left(x \right)}} \]algebraRewrite 1/tan(x) as cos(x)/sin(x).✓ Proved
- \[ = \frac{3}{2 \sin{\left(x \right)} \cos{\left(x \right)}} \]trig algebra algebraUse the identity sec(x) = 1/cos(x). Simplify the expression by canceling cos(x). Multiply numerator and denominator by 2.✓ Proved
- \[ = \frac{3}{\sin{\left(2 x \right)}} \]trig algebraUse the double angle identity sin(2x) = 2sin(x)cos(x). Simplify the expression.✓ Proved
- \[ = 3 \csc{\left(2 x \right)} \]trigUse the identity 1/sin(u) = csc(u).✓ Proved
Answer \( \frac{3}{\sin{\left(2 x \right)}} \)
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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(x) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(x) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(x) = 0 tan has poles at odd multiples of pi/2 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where cos(x)**2 - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(x)**2 - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(x)**2 - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(x)**2 - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(x)**2 - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(x)**2 - 1 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(x)**2 - 1 = 0 undefined where sin(x) = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(x) = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(x) = 0 |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(x) = 0 cot has poles at multiples of pi |
| 20 | ✓ 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 |
| 21 | ✓ 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 tan(x) = 0 |
| 22 | ✓ 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 |
| 23 | ✓ 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 sec has poles at odd multiples of pi/2 |
| 24 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(x) = 0 undefined where sin(x) = 0 |
| 25 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 undefined where sin(x) = 0 undefined where cos(x) = 0 |
| 26 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x) = 0 undefined where sin(x) = 0 |
| 27 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x) = 0 undefined where sin(x) = 0 |
| 28 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x) = 0 undefined where sin(x) = 0 undefined where sin(2*x) = 0 |
| 29 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x) = 0 |
| 30 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*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(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 in single-step increments. The labels accurately reflect the operations performed, and the final result matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — 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: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels used are appropriate for the operations performed, and the final result matches the stated answer.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. The labels used are consistent with the provided vocabulary, and the final result matches the stated answer.gpt-oss:20b: pass 2026-09-20
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.