Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{5} + \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{5} \)
Problem 2.1195 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{5} + \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{5} + \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{5}\right) + \frac{d}{d x} \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{5} \]sum constant-multipleApply the sum rule. Factor out the constant 1/5.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{5} + \frac{\frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{5} \]constant-multipleFactor out the constant -1/5 and 1/5.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(5 x \right)} + 1\right)}{5 \left(\sin{\left(5 x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\sin{\left(5 x \right)} - 1\right)}{5 \left(\sin{\left(5 x \right)} - 1\right)} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{\cos{\left(5 x \right)} \frac{d}{d x} 5 x}{5 \left(\sin{\left(5 x \right)} + 1\right)} - \frac{\cos{\left(5 x \right)} \frac{d}{d x} 5 x}{5 \left(\sin{\left(5 x \right)} - 1\right)} \]chainApply the chain rule to the sine terms.✓ Proved
- \[ = \frac{\cos{\left(5 x \right)}}{\sin{\left(5 x \right)} + 1} - \frac{\cos{\left(5 x \right)}}{\sin{\left(5 x \right)} - 1} \]derivative algebraDifferentiate 5*x. Simplify the constants and coefficients.✓ Proved
- \[ = \left(\frac{1}{\sin{\left(5 x \right)} + 1} - \frac{1}{\sin{\left(5 x \right)} - 1}\right) \cos{\left(5 x \right)} \]algebraFactor out cos(5*x).✓ Proved
- \[ = - \frac{2 \cos{\left(5 x \right)}}{\left(\sin{\left(5 x \right)} - 1\right) \left(\sin{\left(5 x \right)} + 1\right)} \]algebraCombine the fractions using a common denominator.✓ Proved
- \[ = - \frac{2 \cos{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)} - 1} \]algebra algebraSimplify the numerator and denominator. Distribute the cosine term.✓ Proved
- \[ = \frac{2 \cos{\left(5 x \right)}}{1 - \sin^{2}{\left(5 x \right)}} \]algebraSimplify the denominator using the identity 1 - sin^2(x) = cos^2(x).✓ Proved
- \[ = \frac{2}{\cos{\left(5 x \right)}} \]algebra simplifyUse the Pythagorean identity. Cancel the common cosine term.✓ Proved
- \[ = 2 \sec{\left(5 x \right)} \]rewriteRewrite 1/cos(x) as sec(x).✓ Proved
Answer \( \frac{2}{\cos{\left(5 x \right)}} \)
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.
✓ 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(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 6 | ✓ 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 |
| 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 undefined where sin(5*x)**2 - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 undefined where 1 - sin(5*x)**2 = 0 |
| 14 | ✓ 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 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — 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 2026-09-29qwen3.6:27b-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 is sound and the labels are appropriate.gpt-oss:20b: pass 2026-09-29
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.