Home›Calculus 1›Chain rule›Problem 2.220

Derivative of \( \displaystyle - \frac{5 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{5 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \)

Problem 2.220 · hard Beautiful

Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{5 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \).

\[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{5 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4}\right) \]
algebraStart with the derivative of the function. Distribute the division by 4.✓ Proved
\[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4}\right) + \frac{d}{d x} \frac{5 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \]
sumApply the sum rule for derivatives.✓ Proved
\[ = - \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \]
constant-multipleFactor out the constants.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(\sin{\left(2 x \right)} + 1\right)}{4 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{5 \frac{d}{d x} \left(\sin{\left(2 x \right)} - 1\right)}{4 \left(\sin{\left(2 x \right)} - 1\right)} \]
chainApply the chain rule to the logarithmic terms.✓ Proved
\[ = \frac{5 \cos{\left(2 x \right)} \frac{d}{d x} 2 x}{4 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{5 \cos{\left(2 x \right)} \frac{d}{d x} 2 x}{4 \left(\sin{\left(2 x \right)} - 1\right)} \]
chainApply the chain rule to the sine terms.✓ Proved
\[ = \frac{5 \cos{\left(2 x \right)}}{2 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{5 \cos{\left(2 x \right)}}{2 \left(\sin{\left(2 x \right)} - 1\right)} \]
derivative algebraDifferentiate the inner linear function 2*x. Multiply the constants.✓ Proved
\[ = \frac{5 \left(\frac{1}{\sin{\left(2 x \right)} + 1} - \frac{1}{\sin{\left(2 x \right)} - 1}\right) \cos{\left(2 x \right)}}{2} \]
algebraFactor out the common term (5/2) * cos(2*x).✓ Proved
\[ = - \frac{5 \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} - 1\right) \left(\sin{\left(2 x \right)} + 1\right)} \]
algebraFind a common denominator for the terms in the parentheses.✓ Proved
\[ = - \frac{5 \cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} - 1} \]
algebra algebraSimplify the numerator and denominator. Multiply the terms and simplify.✓ Proved
\[ = \frac{5 \cos{\left(2 x \right)}}{1 - \sin^{2}{\left(2 x \right)}} \]
algebraMultiply the numerator and denominator by -1.✓ Proved
\[ = \frac{5}{\cos{\left(2 x \right)}} \]
rewrite simplifyUse the trigonometric identity 1 - sin(x)^2 = cos(x)^2. Cancel the common cos(2*x) term.✓ Proved
\[ = 5 \sec{\left(2 x \right)} \]
rewriteRewrite 1/cos(x) as sec(x).✓ Proved

Answer \( \frac{5}{\cos{\left(2 x \right)}} \)

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
5	✓ 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
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
11	✓ 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
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x)*2 - 1 = 0
13	✓ 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
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - sin(2x)2 = 0 undefined where cos(2x) = 0
15	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*x) = 0
16	✓ 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 6 applies two rules at once: it uses the chain rule on the sine term and simultaneously differentiates the inner linear function 2*x. This violates the requirement that each step change only one rule.
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 6 applies two rules at once: it uses the chain rule on the sine term and simultaneously differentiates the inner linear function 2*x. This violates the requirement that each step change only one rule.
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.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-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 2026-09-19
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 changes only one aspect of the expression, and the labels accurately reflect the operations performed.
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-17
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.