Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6} \)
Problem 2.103 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6}\right) + \frac{d}{d x} \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6} \]sumSplit the derivative into two parts.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{\frac{d}{d x} \ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6} \]constant-multipleFactor out the constant 1/6.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(3 x \right)} + 1\right)}{6 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\sin{\left(3 x \right)} - 1\right)}{6 \left(\sin{\left(3 x \right)} - 1\right)} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{\cos{\left(3 x \right)} \frac{d}{d x} 3 x}{6 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{\cos{\left(3 x \right)} \frac{d}{d x} 3 x}{6 \left(\sin{\left(3 x \right)} - 1\right)} \]chainApply the chain rule to the sine terms.✓ Proved
- \[ = \frac{\cos{\left(3 x \right)}}{2 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{\cos{\left(3 x \right)}}{2 \left(\sin{\left(3 x \right)} - 1\right)} \]derivative algebra simplify algebraDifferentiate the inner function 3*x. Multiply the constants into the numerators. Simplify the fractions. Rearrange the terms to prepare for a common denominator.✓ Proved
- \[ = \frac{\left(\frac{1}{\sin{\left(3 x \right)} + 1} - \frac{1}{\sin{\left(3 x \right)} - 1}\right) \cos{\left(3 x \right)}}{2} \]algebraFactor out cos(3*x).✓ Proved
- \[ = - \frac{\cos{\left(3 x \right)}}{\left(\sin{\left(3 x \right)} - 1\right) \left(\sin{\left(3 x \right)} + 1\right)} \]algebraFind a common denominator for the terms in the parentheses.✓ Proved
- \[ = - \frac{\cos{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)} - 1} \]algebra simplify algebraDistribute the negative sign in the numerator. Simplify the numerator. Simplify the fraction by canceling 2.✓ Proved
- \[ = \frac{\cos{\left(3 x \right)}}{1 - \sin^{2}{\left(3 x \right)}} \]algebraMultiply the numerator and denominator by -1.✓ Proved
- \[ = \frac{1}{\cos{\left(3 x \right)}} \]rewrite simplifyUse the identity 1 - sin(u)**2 = cos(u)**2 with u = 3x. Simplify the fraction by canceling one cos(3*x).✓ Proved
- \[ = \sec{\left(3 x \right)} \]rewriteRewrite 1/cos(3*x) as sec(3*x).✓ Proved
Answer \( \frac{1}{\cos{\left(3 x \right)}} \)
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 |
| 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 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 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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: fail (error) — Step 5 applies two rules at once: the chain rule for the sine function and the derivative of the inner linear function 3*x. The label "chain" only accounts for one of these operations, violating the one‑rule‑per‑step requirement.deepseek-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: fail (error) 2026-09-20 — Step 5 applies two rules at once: the chain rule for the sine function and the derivative of the inner linear function 3*x. The label "chain" only accounts for one of these operations, violating the one‑rule‑per‑step requirement.qwen3.6:27b-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 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed at each stage.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 5 applies two rules at once: it uses the chain rule for sin(3*x) and then separately differentiates 3*x. Each step must change only one thing, so this is a defect.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and trigonometric identities are applied correctly and labeled appropriately.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 14’s note claims a cancellation of 2, yet the expression contains no factor of 2. The rule label is correct, but the explanatory note misleads the student.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.