Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.1097

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

Problem 2.1097 · hard Beautiful

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

\[ \frac{d}{d x} \left(\frac{\ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\ln{\left(\sin{\left(x \right)} + 1 \right)}}{2}\right) \]
Start with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\frac{d}{d x} \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \]
constant-multiple algebraApply the constant multiple rule to each term. Factor out the common constant 1/2.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\sin{\left(x \right)} + 1\right)}{2 \left(\sin{\left(x \right)} + 1\right)} + \frac{\frac{d}{d x} \left(\sin{\left(x \right)} - 1\right)}{2 \left(\sin{\left(x \right)} - 1\right)} \]
chainApply the chain rule to the logarithmic terms.✓ Proved
\[ = - \frac{\cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right)} \]
derivativeDifferentiate the inner functions sin(x) - 1 and sin(x) + 1.✓ Proved
\[ = \frac{\left(- \frac{1}{\sin{\left(x \right)} + 1} + \frac{1}{\sin{\left(x \right)} - 1}\right) \cos{\left(x \right)}}{2} \]
algebraFactor out cos(x).✓ Proved
\[ = \frac{\cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right)} \]
algebraFind a common denominator for the terms in the parentheses.✓ Proved
\[ = \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} - 1} \]
simplify simplifySimplify the numerator and the denominator. Simplify the expression by canceling the 2.✓ Proved
\[ = - \frac{\cos{\left(x \right)}}{1 - \sin^{2}{\left(x \right)}} \]
algebraRewrite the denominator to use the identity 1 - sin(x)**2.✓ Proved
\[ = - \frac{1}{\cos{\left(x \right)}} \]
simplify simplifyUse the trigonometric identity 1 - sin(x)**2 = cos(x)**2. Simplify the fraction by canceling one cos(x).✓ Proved
\[ = - \sec{\left(x \right)} \]
rewriteRewrite the expression using the secant function.✓ Proved

Answer \( - \frac{1}{\cos{\left(x \right)}} \)

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
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 undefined where sin(x) + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 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 undefined where sin(x)**2 - 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 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 undefined where 1 - sin(x)2 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - sin(x)**2 = 0 undefined where cos(x) = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) = 0
13	✓ 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 in single-step increments. The final rewrite to -sec(x) is valid and consistent with the stated answer.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The final rewrite to -sec(x) is valid and consistent with the stated answer.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied.
gpt-oss:20b: pass 2026-09-28

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 2026-09-28 with SymPy 1.14.0.