Derivative of \( \displaystyle - \frac{5 \ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{5 \ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6} \)
Problem 2.1395 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{5 \ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6} \).
- \[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{5 \ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6}\right) \]derivative constantDifferentiate the function with respect to x. Distribute the denominator 6.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6}\right) + \frac{d}{d x} \frac{5 \ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6} \]sumApply the sum rule for derivatives.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6} \]constant-multipleFactor out the constant coefficients.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\sin{\left(3 x \right)} + 1\right)}{6 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{5 \frac{d}{d x} \left(\sin{\left(3 x \right)} - 1\right)}{6 \left(\sin{\left(3 x \right)} - 1\right)} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{5 \left(\frac{d}{d x} 1 + \frac{d}{d x} \sin{\left(3 x \right)}\right)}{6 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{5 \left(- \frac{d}{d x} 1 + \frac{d}{d x} \sin{\left(3 x \right)}\right)}{6 \left(\sin{\left(3 x \right)} - 1\right)} \]sumApply the sum rule inside the derivatives.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sin{\left(3 x \right)}}{6 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{5 \frac{d}{d x} \sin{\left(3 x \right)}}{6 \left(\sin{\left(3 x \right)} - 1\right)} \]constantThe derivative of a constant is zero.✓ Proved
- \[ = \frac{5 \cos{\left(3 x \right)}}{2 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{5 \cos{\left(3 x \right)}}{2 \left(\sin{\left(3 x \right)} - 1\right)} \]trig algebraDifferentiate the sine function. Simplify the coefficients.✓ Proved
- \[ = \frac{5 \left(\frac{1}{\sin{\left(3 x \right)} + 1} - \frac{1}{\sin{\left(3 x \right)} - 1}\right) \cos{\left(3 x \right)}}{2} \]algebraFactor out common terms.✓ Proved
- \[ = - \frac{5 \cos{\left(3 x \right)}}{\left(\sin{\left(3 x \right)} - 1\right) \left(\sin{\left(3 x \right)} + 1\right)} \]algebra algebra algebraFind a common denominator. Simplify the numerator. Simplify the expression.✓ Proved
- \[ = - \frac{5 \cos{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)} - 1} \]algebraExpand the denominator.✓ Proved
- \[ = \frac{5 \cos{\left(3 x \right)}}{1 - \sin^{2}{\left(3 x \right)}} \]algebraMultiply numerator and denominator by -1.✓ Proved
- \[ = \frac{5}{\cos{\left(3 x \right)}} \]trig algebraUse the identity 1 - sin(u)^2 = cos(u)^2. Cancel the common cosine term.✓ Proved
- \[ = 5 \sec{\left(3 x \right)} \]trigRewrite 1/cos(u) as sec(u).✓ Proved
Answer \( \frac{5}{\cos{\left(3 x \right)}} \)
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(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 undefined where sin(3*x)**2 - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x)**2 - 1 = 0 undefined where 1 - sin(3*x)**2 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - sin(3*x)**2 = 0 undefined where cos(3*x) = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x) = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*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(3*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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied.gpt-oss:20b: pass 2026-10-03
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.