Derivative of \( \displaystyle - \frac{5 \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{5 \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \)
Problem 2.1252 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{5 \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{5 \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2}\right) \]derivative algebraStart with the derivative of the function. Distribute the denominator.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2}\right) + \frac{d}{d x} \frac{5 \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \]constant-multipleFactor out the constants.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\sin{\left(x \right)} + 1\right)}{2 \left(\sin{\left(x \right)} + 1\right)} - \frac{5 \frac{d}{d x} \left(\sin{\left(x \right)} - 1\right)}{2 \left(\sin{\left(x \right)} - 1\right)} \]chainApply the chain rule to the logarithms.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sin{\left(x \right)}}{2 \left(\sin{\left(x \right)} + 1\right)} - \frac{5 \frac{d}{d x} \sin{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right)} \]constantDifferentiate the inner term sin(x) - 1.✓ Proved
- \[ = \frac{5 \cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} + 1\right)} - \frac{5 \cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right)} \]trigDifferentiate sin(x) to get cos(x).✓ Proved
- \[ = \frac{5 \cos{\left(x \right)}}{2 \sin{\left(x \right)} + 2} - \frac{5 \cos{\left(x \right)}}{2 \sin{\left(x \right)} - 2} \]algebraSimplify the fractions.✓ Proved
- \[ = \frac{5 \left(\frac{1}{\sin{\left(x \right)} + 1} - \frac{1}{\sin{\left(x \right)} - 1}\right) \cos{\left(x \right)}}{2} \]algebraFactor out the common term (5*cos(x)/2).✓ Proved
- \[ = - \frac{5 \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right)} \]algebraFind a common denominator for the terms in the parentheses.✓ Proved
- \[ = - \frac{5 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} - 1} \]algebraSimplify the numerator and denominator.✓ Proved
- \[ = \frac{5}{\cos{\left(x \right)}} \]algebra algebra algebra simplifyUse the identity sin(x)**2 - 1 = -cos(x)**2. Simplify the signs. Multiply the terms together. Cancel one factor of cos(x).✓ Proved
- \[ = 5 \sec{\left(x \right)} \]rewriteRewrite 1/cos(x) as sec(x).✓ Proved
Answer \( \frac{5}{\cos{\left(x \right)}} \)
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 |
| 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 undefined where 2*sin(x) + 2 = 0 undefined where 2*sin(x) - 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*sin(x) + 2 = 0 undefined where 2*sin(x) - 2 = 0 undefined where sin(x) + 1 = 0 undefined where sin(x) - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) + 1 = 0 undefined where sin(x) - 1 = 0 |
| 11 | ✓ 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 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x)**2 - 1 = 0 undefined where cos(x) = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 |
| 16 | ✓ 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: fail (style) — Step 6 incorrectly labels the differentiation of sin(x)-1 as a "constant" rule; it should be a "derivative" step. No other mathematical errors are present.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 contract.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — 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 contract.gpt-oss:20b: fail (style) 2026-09-29 — Step 6 incorrectly labels the differentiation of sin(x)-1 as a "constant" rule; it should be a "derivative" step. No other mathematical errors are present.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification and trigonometric identities are applied correctly.gpt-oss:20b: fail (style) 2026-09-29 — Step 6 incorrectly labels the operation as "constant"; the step applies the derivative rule to the inner function sin(x) – 1, not a constant‑multiple rule. This mislabeling could mislead a student about which rule is used.
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 2026-09-29 with SymPy 1.14.0.