Derivative of \( \displaystyle \frac{5 \ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{5 \ln{\left(\cos{\left(x \right)} + 1 \right)}}{2} \)
Problem 2.903 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{5 \ln{\left(\cos{\left(x \right)} + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(\frac{5 \ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{5 \ln{\left(\cos{\left(x \right)} + 1 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{5 \frac{d}{d x} \ln{\left(\cos{\left(x \right)} + 1 \right)}}{2} \]sum constant-multipleApply the sum rule for differentiation. Factor out the common constant 5/2.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \left(\cos{\left(x \right)} + 1\right)}{2 \left(\cos{\left(x \right)} + 1\right)} + \frac{5 \frac{d}{d x} \left(\cos{\left(x \right)} - 1\right)}{2 \left(\cos{\left(x \right)} - 1\right)} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \cos{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)} + \frac{5 \frac{d}{d x} \cos{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)} \]sumDifferentiate the terms inside the parentheses.✓ Proved
- \[ = \frac{5 \sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)} - \frac{5 \sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)} \]trig algebraThe derivative of cos(x) is -sin(x). Simplify the signs and distribute the terms.✓ Proved
- \[ = \frac{5 \left(\frac{1}{\cos{\left(x \right)} + 1} - \frac{1}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2} \]algebraFactor out sin(x).✓ Proved
- \[ = - \frac{5 \sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)} \]algebraCombine the fractions using a common denominator.✓ Proved
- \[ = - \frac{5 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 1} \]algebra algebraExpand the numerator and denominator. Simplify the numerator.✓ Proved
- \[ = \frac{5 \sin{\left(x \right)}}{1 - \cos^{2}{\left(x \right)}} \]algebraDistribute the negative sign into the denominator.✓ Proved
- \[ = \frac{5}{\sin{\left(x \right)}} \]trig simplifyUse the identity 1 - cos(x)**2 = sin(x)**2. Simplify the expression by canceling terms.✓ Proved
Answer \( \frac{5}{\sin{\left(x \right)}} \)
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.
✓ 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 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 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x)**2 - 1 = 0 undefined where 1 - cos(x)**2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - cos(x)**2 = 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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (style) 2026-09-26 — Step 1 is labeled "unlabelled" even though it applies the derivative rule. The label should be "derivative" to match the allowed vocabulary.
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.