Home›Calculus 1›Chain rule›Problem 2.260

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

Problem 2.260 · hard Beautiful

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

\[ \frac{d}{d x} \left(\frac{3 \ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{3 \ln{\left(\cos{\left(x \right)} + 1 \right)}}{2}\right) \]
constantStart with the derivative of the function.✓ Proved
\[ = \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(x \right)} + 1 \right)}}{2} \]
sum algebraApply the linearity of the derivative. Factor out the common constant 3/2.✓ Proved
\[ = - \frac{3 \frac{d}{d x} \left(\cos{\left(x \right)} + 1\right)}{2 \left(\cos{\left(x \right)} + 1\right)} + \frac{3 \frac{d}{d x} \left(\cos{\left(x \right)} - 1\right)}{2 \left(\cos{\left(x \right)} - 1\right)} \]
logarithmicApply the chain rule for the natural logarithm.✓ Proved
\[ = - \frac{3 \frac{d}{d x} \cos{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)} + \frac{3 \frac{d}{d x} \cos{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)} \]
constantThe derivative of the constant -1 and 1 is zero.✓ Proved
\[ = \frac{3 \left(- \frac{1}{\cos{\left(x \right)} + 1} + \frac{1}{\cos{\left(x \right)} - 1}\right) \frac{d}{d x} \cos{\left(x \right)}}{2} \]
algebraFactor out the common derivative term.✓ Proved
\[ = - \frac{3 \left(- \frac{1}{\cos{\left(x \right)} + 1} + \frac{1}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2} \]
trigDifferentiate the cosine function.✓ Proved
\[ = - \frac{3 \sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)} \]
algebra algebraFind a common denominator for the terms in the parentheses. Simplify the numerator in the parentheses.✓ Proved
\[ = - \frac{3 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 1} \]
algebraExpand the denominator using the difference of squares.✓ Proved
\[ = \frac{3}{\sin{\left(x \right)}} \]
algebraUse the identity cos(x)**2 - 1 = -sin(x)**2.✓ Proved
\[ = - \frac{3 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 1} \]
algebraMultiply the terms together.✓ Proved
\[ = \frac{3}{\sin{\left(x \right)}} \]
algebra simplifySubstitute the denominator back. Simplify the fraction by canceling sin(x) and handling the signs.✓ Proved

Answer \( \frac{3}{\sin{\left(x \right)}} \)

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.

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 cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0 undefined where cos(x)**2 - 1 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x)**2 - 1 = 0 undefined where sin(x) = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) = 0 undefined where cos(x)**2 - 1 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x)**2 - 1 = 0 undefined where sin(x) = 0
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(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: pass

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
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: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 12 incorrectly multiplies the terms: 3/2*(-sin(x))*(2/(-sin(x)**2)) simplifies to 3/sin(x), not -3*sin(x)/(cos(x)**2-1).
gpt-oss:20b: fail 2026-09-17 — The function f(x)=3/2·log(cos x−1)−3/2·log(cos x+1) is not real‑valued for any real x (cos x−1≤0 and cos x+1≥0, with zeros at multiples of π). Thus the derivative 3/ sin x is not valid on the real domain, and the solution ignores this domain/branch‑cut restriction.
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.