Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{3} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{3} \)
Problem 2.147 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{3} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{3} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{3}\right) \]sumApply the sum rule to differentiate each term separately.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{3}\right) + \frac{d}{d x} \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{3} \]constant-multipleFactor out the constant 1/3 from each term.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{3} + \frac{\frac{d}{d x} \ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{3} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(3 x \right)} + 1\right)}{3 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\sin{\left(3 x \right)} - 1\right)}{3 \left(\sin{\left(3 x \right)} - 1\right)} \]logarithmicDifferentiate the natural logarithm functions.✓ Proved
- \[ = \frac{\frac{d}{d x} \sin{\left(3 x \right)}}{3 \left(\sin{\left(3 x \right)} + 1\right)} - \frac{\frac{d}{d x} \sin{\left(3 x \right)}}{3 \left(\sin{\left(3 x \right)} - 1\right)} \]constantDifferentiate the constant -1 and 1 inside the derivatives.✓ Proved
- \[ = \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)} + 1} - \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)} - 1} \]trig algebraDifferentiate the sine functions using the chain rule. Simplify the coefficients and products.✓ Proved
- \[ = \left(\frac{1}{\sin{\left(3 x \right)} + 1} - \frac{1}{\sin{\left(3 x \right)} - 1}\right) \cos{\left(3 x \right)} \]algebraFactor out the common term cos(3*x).✓ Proved
- \[ = - \frac{2 \cos{\left(3 x \right)}}{\left(\sin{\left(3 x \right)} - 1\right) \left(\sin{\left(3 x \right)} + 1\right)} \]algebraCombine the fractions using a common denominator.✓ Proved
- \[ = - \frac{2 \cos{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)} - 1} \]algebra simplifySimplify the numerator and denominator. Final simplification.✓ Proved
- \[ = \frac{2 \cos{\left(3 x \right)}}{1 - \sin^{2}{\left(3 x \right)}} \]simplifyUse the identity 1 - sin(x)^2 = cos(x)^2 to simplify further.✓ Proved
- \[ = \frac{2}{\cos{\left(3 x \right)}} \]simplify simplifySubstitute the trigonometric identity. Cancel the common cosine term.✓ Proved
- \[ = 2 \sec{\left(3 x \right)} \]rewriteRewrite the expression using the secant function.✓ Proved
Answer \( \frac{2}{\cos{\left(3 x \right)}} \)
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 undefined where 1 - sin(3*x)**2 = 0 |
| 13 | ✓ 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 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 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 (style) — Step 3 is labeled "chain" but it only applies the constant‑multiple rule to pull out the factor –1/3; the chain rule is not invoked at that point. This mislabeling violates the one‑rule‑per‑step requirement.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'chain' but performs no differentiation; it merely prepares for the chain rule. The actual differentiation of the logarithm occurs in Step 4, which is correctly labeled 'logarithmic'. Labeling a non-differentiation step as 'chain' is incorrect.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'chain' but performs no differentiation; it merely prepares for the chain rule. The actual differentiation of the logarithm occurs in Step 4, which is correctly labeled 'logarithmic'. Labeling a non-differentiation step as 'chain' is incorrect.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 is labeled "chain" but it only applies the constant‑multiple rule to pull out the factor –1/3; the chain rule is not invoked at that point. This mislabeling violates the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'chain' but performs no differentiation; it merely prepares for the chain rule by isolating the log terms. The actual application of the chain rule (differentiating the outer log function) occurs in Step 4, which is correctly labeled 'logarithmic'. Labeling Step 3 as 'chain' is incorrect because the chain rule was not applied in that step.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule; it only pulls out the constant factor. Step 6 labels the derivative of sin(3*x) as a trig rule, but it is actually a chain rule application. Both steps violate the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 is labeled 'chain' but performs no differentiation; it merely prepares the expression for the chain rule in Step 4. Step 6 is labeled 'trig' but applies the chain rule to differentiate sin(3x), which should be labeled 'chain' or 'derivative'. Furthermore, Step 3 violates the one-rule-per-step constraint by implicitly handling the structure for the next step without performing the actual differentiation labeled in Step 4.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 is labeled 'chain' but performs no differentiation; it merely rearranges terms, which should be labeled 'constant-multiple' or 'algebra'. Step 4 is labeled 'logarithmic' but applies the chain rule structure (introducing Derivative(...)), which should be labeled 'chain'. The labels are swapped.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: inconclusive 2026-09-19 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 5 incorrectly applies the \"constant\" rule to the inner derivatives. It omits the fact that the derivative of the constant terms \(-1\) and \(+1\) iqwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 3 is labeled 'chain' but performs no differentiation; it merely sets up the structure for the next step. The actual differentiation of the logarithm occurs in Step 4, which is correctly labeled 'logarithmic'. Labeling Step 3 as 'chain' is incorrect because the chain rule is not applied until the inner derivative is taken in subsequent steps, and Step 3 itself is just a structural rewrite or preparation, not a rule application that changes the mathematical value via differentiation. Furthermore, Step 6 claims to use the 'trig' rule to differentiate sine, but the differentiation of sin(3x) requires the chain rule (derivative of inner function 3x), which is not explicitly labeled as such in a single step, though Step 6 combines the derivative of sin and the constant multiple from the inner function. However, the primary defect is Step 3: it claims to apply the chain rule but does not compute any derivative, it just writes out the form. The chain rule application is split across steps 3, 4, and 6 in a confusing way. Specifically, Step 3 says 'Apply the chain rule' but the expression is unchanged in terms of differentiation progress; it's just a placeholder. A step labeled 'chain' should result in the derivative of the outer function times the derivative of the inner function, or at least the derivative of the outer function. Step 3 does neither; it just writes 'Derivative(log(...))'. This is not an application of the chain rule. It's a no-op or a rewrite. The label is wrong.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (error) 2026-09-18 — Step 3 incorrectly labels the application of the logarithmic derivative as "chain"; it should be "logarithmic". Step 5 also mislabels the differentiation of the constants inside the arguments: the derivative of "+1" and "-1" is zero and should be omitted, yet the step claims a "constant" rule was applied. These mislabelings and omissions make the solution mathematically incorrect.gpt-oss:20b: fail 2026-09-17 — Step 3 incorrectly labels the rule as chain; it only applies the constant‑multiple rule, not the chain rule.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.