Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \)
Problem 2.83 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \).
- \[ \frac{d}{d x} \left(\frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10}\right) \]sumApply the linearity of the derivative.✓ Proved
- \[ = \frac{d}{d x} \frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{d}{d x} \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \]constant-multipleFactor out the constant 1/10 from each term.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{\frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cos{\left(5 x \right)} + 1\right)}{10 \left(\cos{\left(5 x \right)} + 1\right)} + \frac{\frac{d}{d x} \left(\cos{\left(5 x \right)} - 1\right)}{10 \left(\cos{\left(5 x \right)} - 1\right)} \]chainApply the chain rule to the inner functions.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(5 x \right)}}{10 \left(\cos{\left(5 x \right)} + 1\right)} + \frac{\frac{d}{d x} \cos{\left(5 x \right)}}{10 \left(\cos{\left(5 x \right)} - 1\right)} \]constantThe derivative of the constant -1 and +1 is zero.✓ Proved
- \[ = \frac{\sin{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} + 1\right)} - \frac{\sin{\left(5 x \right)}}{2 \left(\cos{\left(5 x \right)} - 1\right)} \]trig algebra simplifyDifferentiate the cosine function. Simplify the constants and signs. Simplify the fraction 5/10.✓ Proved
- \[ = \frac{\frac{\left(\cos{\left(5 x \right)} - 1\right) \sin{\left(5 x \right)}}{2} - \frac{\left(\cos{\left(5 x \right)} + 1\right) \sin{\left(5 x \right)}}{2}}{\left(\cos{\left(5 x \right)} - 1\right) \left(\cos{\left(5 x \right)} + 1\right)} \]algebraFind a common denominator.✓ Proved
- \[ = - \frac{\sin{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)} - 1} \]algebra simplify simplifyDistribute the sine term in the numerator. Combine like terms in the numerator. Simplify the fraction.✓ Proved
- \[ = \frac{\sin{\left(5 x \right)}}{1 - \cos^{2}{\left(5 x \right)}} \]algebraMultiply numerator and denominator by -1.✓ Proved
- \[ = \frac{1}{\sin{\left(5 x \right)}} \]trig simplifyUse the Pythagorean identity 1 - cos^2(u) = sin^2(u). Simplify the fraction.✓ Proved
- \[ = \csc{\left(5 x \right)} \]rewriteRewrite 1/sin(x) as csc(x).✓ Proved
Answer \( \frac{1}{\sin{\left(5 x \right)}} \)
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(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) + 1 = 0 undefined where cos(5*x) - 1 = 0 undefined where cos(5*x)**2 - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x)**2 - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x)**2 - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x)**2 - 1 = 0 undefined where 1 - cos(5*x)**2 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - cos(5*x)**2 = 0 undefined where sin(5*x) = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) = 0 csc has poles at multiples of pi |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(5*x) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (style) — Step 2 applies the constant-multiple rule to both terms simultaneously, violating the one-change-per-step constraint. Step 3 labels the application of the chain rule for logarithms as 'logarithmic', but the actual differentiation of the inner function occurs in Step 4; Step 3 is better described as 'chain' or 'rewrite' if it's just setting up the structure, but given the label 'logarithmic' usually implies d/dx log(u) = u'/u, it conflates the rule with the next step's execution.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 applies the constant-multiple rule to both terms simultaneously, violating the one-change-per-step constraint. Step 3 labels the application of the chain rule for logarithms as 'logarithmic', but the actual differentiation of the inner function occurs in Step 4; Step 3 is better described as 'chain' or 'rewrite' if it's just setting up the structure, but given the label 'logarithmic' usually implies d/dx log(u) = u'/u, it conflates the rule with the next step's execution.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-rule constraint, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 2 applies the constant multiple rule to both terms simultaneously, violating the one-change-per-step constraint. Step 3 incorrectly labels the application of the chain rule for logarithms as 'logarithmic'; the 'logarithmic' rule typically refers to the basic derivative of log(x), whereas the chain rule application is explicitly labeled in Step 4.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 2 incorrectly labels the application of the sum rule as "constant‑multiple". Step 3 incorrectly labels the extraction of the constant 1/10 as "logarithmic"; it should be "constant‑multiple". These labeling errors violate the contract on rule granularity.qwen3.6:27b-mlx: inconclusive 2026-09-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 3 is labeled 'logarithmic' but performs no differentiation; it merely sets up the chain rule application which occurs in Step 4. The label 'logdeepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 2 is mislabelled: it applies the linearity rule, not a constant‑multiple rule, and it does not yet factor out the 1/10. Step 3 is also mislabelled: it applies the constant‑multiple rule, not a logarithmic rule.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.