Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.988

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

Problem 2.988 · hard Beautiful

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

\[ \frac{d}{d x} \left(- \ln{\left(\sin{\left(x \right)} - 1 \right)} + \ln{\left(\sin{\left(x \right)} + 1 \right)}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\sin{\left(x \right)} - 1 \right)} + \frac{d}{d x} \ln{\left(\sin{\left(x \right)} + 1 \right)} \]
sumApply the sum rule.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\sin{\left(x \right)} - 1 \right)} + \frac{\frac{d}{d x} \left(\sin{\left(x \right)} + 1\right)}{\sin{\left(x \right)} + 1} \]
chainApply the chain rule to the second term.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\sin{\left(x \right)} - 1 \right)} + \frac{\frac{d}{d x} \sin{\left(x \right)}}{\sin{\left(x \right)} + 1} \]
constantThe derivative of the constant 1 is 0.✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(x \right)}}{\sin{\left(x \right)} + 1} - \frac{\frac{d}{d x} \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} \]
chainApply the chain rule to the first term.✓ Proved
\[ = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1} - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)} - 1} \]
trigDifferentiate sin(x) to get cos(x).✓ Proved
\[ = \left(\frac{1}{\sin{\left(x \right)} + 1} - \frac{1}{\sin{\left(x \right)} - 1}\right) \cos{\left(x \right)} \]
algebraFactor out cos(x).✓ Proved
\[ = - \frac{2 \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right)} \]
algebraFind a common denominator.✓ Proved
\[ = - \frac{2 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} - 1} \]
algebra simplify algebraSimplify the numerator and denominator. Simplify the numerator. Multiply the terms.✓ Proved
\[ = \frac{2 \cos{\left(x \right)}}{1 - \sin^{2}{\left(x \right)}} \]
algebraDistribute the negative sign into the denominator.✓ Proved
\[ = \frac{2}{\cos{\left(x \right)}} \]
algebra simplifyUse the identity 1 - sin(x)**2 = cos(x)**2. Simplify the fraction.✓ Proved
\[ = 2 \sec{\left(x \right)} \]
trigRewrite 1/cos(x) as sec(x).✓ Proved

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

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.

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.

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 undefined where sin(x) + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(x) + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(x) + 1 = 0 undefined where sin(x) - 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) + 1 = 0 undefined where sin(x) - 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) + 1 = 0 undefined where sin(x) - 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) + 1 = 0 undefined where sin(x) - 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) + 1 = 0 undefined where sin(x) - 1 = 0 undefined where sin(x)**2 - 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x)**2 - 1 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x)**2 - 1 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x)2 - 1 = 0 undefined where 1 - sin(x)2 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - sin(x)**2 = 0 undefined where cos(x) = 0
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) = 0
15	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(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(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
qwen3.6:27b-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 (4)

qwen3.6:27b-mlx: pass 2026-09-27 — 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: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the sum, chain, and algebraic simplification rules in single-step increments. The final answer matches the stated answer, and the steps are logically sound.
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.