Home›Calculus 1›Chain rule›Problem 2.108

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

Problem 2.108 · hard Beautiful

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

\[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{2}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} \frac{\ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{2} + \frac{d}{d x} \frac{\ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{2} \]
sumApply the sum rule.✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{2} \]
constant-multipleFactor out the constant 1/2.✓ Proved
\[ = \frac{\frac{d}{d x} \left(\sin{\left(2 x \right)} + 1\right)}{2 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\sin{\left(2 x \right)} - 1\right)}{2 \left(\sin{\left(2 x \right)} - 1\right)} \]
chainApply the chain rule to both logarithmic terms.✓ Proved
\[ = \frac{\cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} - \frac{\cos{\left(2 x \right)}}{\sin{\left(2 x \right)} - 1} \]
chain algebraApply the chain rule to the sine terms. Simplify the constants and products.✓ Proved
\[ = \left(\frac{1}{\sin{\left(2 x \right)} + 1} - \frac{1}{\sin{\left(2 x \right)} - 1}\right) \cos{\left(2 x \right)} \]
algebraFactor out cos(2*x).✓ Proved
\[ = - \frac{2 \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} - 1\right) \left(\sin{\left(2 x \right)} + 1\right)} \]
algebraCombine the fractions using a common denominator.✓ Proved
\[ = - \frac{2 \cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} - 1} \]
algebra algebraSimplify the numerator and denominator. Multiply the terms.✓ Proved
\[ = \frac{2 \cos{\left(2 x \right)}}{1 - \sin^{2}{\left(2 x \right)}} \]
algebraDistribute the negative sign into the denominator.✓ Proved
\[ = \frac{2}{\cos{\left(2 x \right)}} \]
rewrite simplifyUse the trigonometric identity 1 - sin(u)^2 = cos(u)^2. Cancel the common cos(2*x) term.✓ Proved
\[ = 2 \sec{\left(2 x \right)} \]
rewriteRewrite 1/cos(u) as sec(u).✓ Proved

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

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 undefined where sin(2x)*2 - 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 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 undefined where 1 - sin(2x)**2 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - sin(2x)2 = 0 undefined where cos(2x) = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*x) = 0
14	✓ 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: fail (error) — Step 4 applies the chain rule to both logarithmic terms in one line, changing two subexpressions at once. The same occurs in step 5, violating the rule that each step must alter only one part of the expression.
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass — 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.

Every verdict on record (15)

qwen3.6:27b-mlx: pass 2026-09-20 — 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.
gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies the chain rule to both logarithmic terms in one line, changing two subexpressions at once. The same occurs in step 5, violating the rule that each step must alter only one part of the expression.
qwen3.6:27b-mlx: pass 2026-09-20 — 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.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (style) 2026-09-19 — Step 4 applies the chain rule to two separate logarithmic terms in a single step, violating the rule that each step must change only one thing. The same issue occurs in step 5.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 is missing a rule label, which is required by the contract.
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules and algebraic simplifications. Each step modifies the expression using a single rule from the allowed vocabulary, and the labels accurately describe the operations performed.
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
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.