Derivative of \( \displaystyle - \frac{3 \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{3 \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \)
Problem 2.542 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{3 \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{3 \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{3 \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2}\right) \]derivative algebraStart with the derivative of the function. Rewrite the coefficients for clarity.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{3 \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2}\right) + \frac{d}{d x} \frac{3 \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{3 \frac{d}{d x} \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{3 \frac{d}{d x} \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\sin{\left(x \right)} + 1\right)}{2 \left(\sin{\left(x \right)} + 1\right)} - \frac{3 \frac{d}{d x} \left(\sin{\left(x \right)} - 1\right)}{2 \left(\sin{\left(x \right)} - 1\right)} \]chainApply the chain rule to each term.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \sin{\left(x \right)}}{2 \left(\sin{\left(x \right)} + 1\right)} - \frac{3 \frac{d}{d x} \sin{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right)} \]sumDifferentiate the inner sum.✓ Proved
- \[ = \frac{3 \cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} + 1\right)} - \frac{3 \cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right)} \]trigDifferentiate the sine function.✓ Proved
- \[ = \frac{3 \cos{\left(x \right)}}{2 \sin{\left(x \right)} + 2} - \frac{3 \cos{\left(x \right)}}{2 \sin{\left(x \right)} - 2} \]algebraCombine the fractions.✓ Proved
- \[ = \frac{3 \left(\frac{1}{\sin{\left(x \right)} + 1} - \frac{1}{\sin{\left(x \right)} - 1}\right) \cos{\left(x \right)}}{2} \]algebraFactor out common terms.✓ Proved
- \[ = - \frac{3 \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right)} \]algebraFind a common denominator.✓ Proved
- \[ = - \frac{3 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} - 1} \]algebra algebraSimplify the numerator. Cancel the common factor of 2.✓ Proved
- \[ = \frac{3 \cos{\left(x \right)}}{1 - \sin^{2}{\left(x \right)}} \]algebraDistribute the negative sign.✓ Proved
- \[ = \frac{3}{\cos{\left(x \right)}} \]trig simplifyUse the identity 1 - sin(x)^2 = cos(x)^2. Cancel one cos(x) term.✓ Proved
- \[ = 3 \sec{\left(x \right)} \]trigRewrite 1/cos(x) as sec(x).✓ Proved
Answer \( \frac{3}{\cos{\left(x \right)}} \)
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 sin(x) - 1 = 0 undefined where sin(x) + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) - 1 = 0 undefined where sin(x) + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) - 1 = 0 undefined where sin(x) + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) - 1 = 0 undefined where sin(x) + 1 = 0 undefined where 2*sin(x) - 2 = 0 undefined where 2*sin(x) + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*sin(x) - 2 = 0 undefined where 2*sin(x) + 2 = 0 undefined where sin(x) - 1 = 0 undefined where sin(x) + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) - 1 = 0 undefined where sin(x) + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) - 1 = 0 undefined where sin(x) + 1 = 0 undefined where sin(x)**2 - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x)**2 - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x)**2 - 1 = 0 undefined where 1 - sin(x)**2 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - sin(x)**2 = 0 undefined where cos(x) = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x) = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x) = 0 sec has poles at odd multiples of pi/2 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where cos(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 changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.