Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \)
Problem 2.127 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10}\right) + \frac{d}{d x} \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \]sum constant-multipleApply the sum rule. Factor out the constant 1/10.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10}\right) - \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10}\right) \]algebraRewrite the second term to facilitate simplification.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{\frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \]algebraDistribute the negative sign.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(5 x \right)} + 1\right)}{10 \left(\sin{\left(5 x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\sin{\left(5 x \right)} - 1\right)}{10 \left(\sin{\left(5 x \right)} - 1\right)} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{\cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} + 1\right)} - \frac{\cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} - 1\right)} \]chain algebra simplifyApply the chain rule to the sine terms. Simplify the constants and products. Simplify the fractions.✓ Proved
- \[ = \frac{\left(\frac{1}{\sin{\left(5 x \right)} + 1} - \frac{1}{\sin{\left(5 x \right)} - 1}\right) \cos{\left(5 x \right)}}{2} \]algebraFactor out common terms.✓ Proved
- \[ = - \frac{\cos{\left(5 x \right)}}{\left(\sin{\left(5 x \right)} - 1\right) \left(\sin{\left(5 x \right)} + 1\right)} \]algebraCombine the fractions using a common denominator.✓ Proved
- \[ = - \frac{\cos{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)} - 1} \]simplify algebraSimplify the numerator and denominator. Cancel the common factor of 2.✓ Proved
- \[ = \frac{\cos{\left(5 x \right)}}{1 - \sin^{2}{\left(5 x \right)}} \]algebraDistribute the negative sign into the denominator.✓ Proved
- \[ = \frac{1}{\cos{\left(5 x \right)}} \]trig simplifyUse the Pythagorean identity 1 - sin^2(u) = cos^2(u). Simplify the fraction.✓ Proved
- \[ = \sec{\left(5 x \right)} \]rewriteRewrite 1/cos(x) as sec(x).✓ Proved
Answer \( \frac{1}{\cos{\left(5 x \right)}} \)
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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 undefined where sin(5*x)**2 - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x)**2 - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x)**2 - 1 = 0 undefined where 1 - sin(5*x)**2 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - sin(5*x)**2 = 0 undefined where cos(5*x) = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*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(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: fail (style) — Step 1 is missing a rule label, which is required by the contract.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.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 isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 is missing a rule label, which is required by the contract.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 4 incorrectly rewrites the second derivative term, changing a positive 1/10 log(sin(5*x)+1) into a negative –1/10 log(sin(5*x)+1). This applies two changes (sign flip and constant factor) in one step, violating the one‑change rule and producing an incorrect expression.qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.deepseek-r1:70b: fail (style) 2026-09-18 — Step 1 lacks a rule label, which could confuse students about the applied rule.gpt-oss:20b: fail (error) 2026-09-18 — Step 4 incorrectly changes the sign of the second derivative term; it should remain +Derivative(1/10*log(sin(5*x)+1),x). Additionally, step 13 claims to cancel a factor of 2, but no such factor exists after step 12, making the algebraic manipulation incorrect.gpt-oss:20b: inconclusive 2026-09-17 — reviewer response could not be parsed: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.