Derivative of \( \displaystyle - \frac{3 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{3 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \)

Problem 2.97 · hard Beautiful

Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{3 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \).

\[ \frac{d}{d x} \left(- \frac{3 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{3 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4}\right) \]
sumStart with the derivative of the entire function.✓ Proved
\[ = \frac{d}{d x} \left(- \frac{3 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4}\right) + \frac{d}{d x} \frac{3 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \]
constant-multipleDistribute the derivative across the sum and pull out constants.✓ Proved
\[ = - \frac{3 \frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{3 \frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \]
constant-multipleGroup the constants for each term.✓ Proved
\[ = \frac{3 \frac{d}{d x} \left(\sin{\left(2 x \right)} + 1\right)}{4 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{3 \frac{d}{d x} \left(\sin{\left(2 x \right)} - 1\right)}{4 \left(\sin{\left(2 x \right)} - 1\right)} \]
logarithmicApply the chain rule for the natural logarithm.✓ Proved
\[ = \frac{3 \frac{d}{d x} \sin{\left(2 x \right)}}{4 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{3 \frac{d}{d x} \sin{\left(2 x \right)}}{4 \left(\sin{\left(2 x \right)} - 1\right)} \]
derivativeDifferentiate the inner terms of the logarithm.✓ Proved
\[ = \frac{3 \cos{\left(2 x \right)}}{2 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{3 \cos{\left(2 x \right)}}{2 \left(\sin{\left(2 x \right)} - 1\right)} \]
chain algebraApply the chain rule to the sine function. Simplify the constants and the products.✓ Proved
\[ = \frac{3 \left(\frac{1}{\sin{\left(2 x \right)} + 1} - \frac{1}{\sin{\left(2 x \right)} - 1}\right) \cos{\left(2 x \right)}}{2} \]
algebraFactor out the common term 3/2 * cos(2*x).✓ Proved
\[ = - \frac{3 \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} - 1\right) \left(\sin{\left(2 x \right)} + 1\right)} \]
algebraCombine the fractions using a common denominator.✓ Proved
\[ = - \frac{3 \cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} - 1} \]
algebra algebraSimplify the numerator and denominator. Multiply the terms and simplify the expression.✓ Proved
\[ = \frac{3 \cos{\left(2 x \right)}}{1 - \sin^{2}{\left(2 x \right)}} \]
algebraDistribute the negative sign into the denominator.✓ Proved
\[ = \frac{3}{\cos{\left(2 x \right)}} \]
rewrite simplifyUse the trigonometric identity 1 - sin^2(u) = cos^2(u). Cancel the common cosine term.✓ Proved
\[ = 3 \sec{\left(2 x \right)} \]
rewriteRewrite the reciprocal of cosine as secant.✓ Proved

Answer \( \frac{3}{\cos{\left(2 x \right)}} \)

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(2x) - 1 = 0 undefined where sin(2x) + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x) - 1 = 0 undefined where sin(2x) + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x) - 1 = 0 undefined where sin(2x) + 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x) - 1 = 0 undefined where sin(2x) + 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x) - 1 = 0 undefined where sin(2x) + 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x) - 1 = 0 undefined where sin(2x) + 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x) - 1 = 0 undefined where sin(2x) + 1 = 0 undefined where sin(2x)*2 - 1 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x)*2 - 1 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x)2 - 1 = 0 undefined where 1 - sin(2x)**2 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - sin(2x)2 = 0 undefined where cos(2x) = 0
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*x) = 0
15	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*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(2*x) = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: fail (error) — Step 2 applies both the sum rule and the constant multiple rule simultaneously, violating the constraint that each step must change only one thing. Step 4 is labeled 'logarithmic' but the note claims it applies the chain rule; while the label is acceptable for the log derivative, the note is misleading about the specific rule application (the chain rule is explicitly applied in subsequent steps).

Every verdict on record (12)

qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 applies both the sum rule and the constant multiple rule simultaneously, violating the constraint that each step must change only one thing. Step 4 is labeled 'logarithmic' but the note claims it applies the chain rule; while the label is acceptable for the log derivative, the note is misleading about the specific rule application (the chain rule is explicitly applied in subsequent steps).
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 applies both the sum rule and the constant-multiple rule simultaneously, violating the constraint that each step must change only one thing. Step 2 should only apply the sum rule, leaving the constants inside the derivative operators for a subsequent step.
gpt-oss:20b: fail (error) 2026-09-20 — Step 5 incorrectly drops the derivative of the constant term –1. The derivative of sin(2*x)-1 should be 2*cos(2*x), not just cos(2*x).
qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 2 applies both the sum rule and the constant-multiple rule simultaneously, violating the one-rule-per-step constraint. Step 4 is labeled 'logarithmic' but the note describes the chain rule; while the label is acceptable for the log derivative, the note is misleading regarding the specific rule applied in that step versus the subsequent chain rule steps.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 applies both the sum rule and the constant multiple rule simultaneously, violating the one-rule-per-step constraint. Step 4 is labeled 'logarithmic' but the note describes the chain rule; while the label is acceptable for the log derivative, the step also implicitly handles the chain rule structure which is usually separated, but the primary defect is Step 2.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: inconclusive 2026-09-17 — reviewer response could not be parsed:
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.