Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{2} \)
Problem 2.204 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{2}\right) + \frac{d}{d x} \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{2} \]constant-multipleFactor out the constant 1/2.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(3 x \right)} + 1\right)}{2 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\sin{\left(3 x \right)} - 1\right)}{2 \left(\sin{\left(3 x \right)} - 1\right)} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = \frac{\cos{\left(3 x \right)} \frac{d}{d x} 3 x}{2 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{\cos{\left(3 x \right)} \frac{d}{d x} 3 x}{2 \left(\sin{\left(3 x \right)} - 1\right)} \]chainApply the chain rule to the sine functions.✓ Proved
- \[ = \frac{3 \cos{\left(3 x \right)}}{2 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{3 \cos{\left(3 x \right)}}{2 \left(\sin{\left(3 x \right)} - 1\right)} \]derivative algebraDifferentiate the inner function 3*x. Simplify the coefficients and terms.✓ Proved
- \[ = \frac{3 \left(\frac{1}{\sin{\left(3 x \right)} + 1} - \frac{1}{\sin{\left(3 x \right)} - 1}\right) \cos{\left(3 x \right)}}{2} \]constant-multipleFactor out the common term 3/2 * cos(3*x).✓ Proved
- \[ = - \frac{3 \cos{\left(3 x \right)}}{\left(\sin{\left(3 x \right)} - 1\right) \left(\sin{\left(3 x \right)} + 1\right)} \]algebraCombine the fractions using a common denominator.✓ Proved
- \[ = - \frac{3 \cos{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)} - 1} \]algebra simplify algebraExpand the numerator and denominator. Simplify the numerator. Multiply the terms and simplify the fraction.✓ Proved
- \[ = \frac{3 \cos{\left(3 x \right)}}{1 - \sin^{2}{\left(3 x \right)}} \]algebraMultiply numerator and denominator by -1.✓ Proved
- \[ = \frac{3}{\cos{\left(3 x \right)}} \]rewrite simplifyUse the trigonometric identity 1 - sin^2(u) = cos^2(u). Cancel the common cos(3*x) term.✓ Proved
- \[ = 3 \sec{\left(3 x \right)} \]rewriteRewrite 1/cos(u) as sec(u).✓ Proved
Answer \( \frac{3}{\cos{\left(3 x \right)}} \)
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.
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 sin(3*x) - 1 = 0 undefined where sin(3*x) + 1 = 0 |
| 5 | ✓ 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 |
| 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 undefined where sin(3*x)**2 - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x)**2 - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x)**2 - 1 = 0 |
| 13 | ✓ 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 |
| 14 | ✓ 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 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x) = 0 |
| 16 | ✓ 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — 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 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels used are from the allowed vocabulary and accurately describe the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 5 applies more than one rule at once: it uses the chain rule for the sine function and simultaneously differentiates the inner function 3*x (and the constant -1). Each step must change only one thing, so this step is a defect.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 8 incorrectly labels the operation as a constant‑multiple rule; it actually factors out a common factor 3/2 cos(3x), where cos(3x) is not a constant. This mislabeling could mislead a student about which rule applies.deepseek-r1:70b: pass 2026-09-17
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.