Derivative of \( \displaystyle - \frac{3 \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{3 \ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \)
Problem 2.284 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{3 \ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \).
- \[ \frac{d}{d x} \left(- \frac{3 \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{3 \ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10}\right) \]constantStart with the derivative of the function.✓ Proved
- \[ = - \frac{3 \frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{3 \frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \]constant-multiplePull out the constant coefficients.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\sin{\left(5 x \right)} + 1\right)}{10 \left(\sin{\left(5 x \right)} + 1\right)} - \frac{3 \frac{d}{d x} \left(\sin{\left(5 x \right)} - 1\right)}{10 \left(\sin{\left(5 x \right)} - 1\right)} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{3 \cos{\left(5 x \right)} \frac{d}{d x} 5 x}{10 \left(\sin{\left(5 x \right)} + 1\right)} - \frac{3 \cos{\left(5 x \right)} \frac{d}{d x} 5 x}{10 \left(\sin{\left(5 x \right)} - 1\right)} \]chainApply the chain rule to the sine terms.✓ Proved
- \[ = \frac{3 \cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} + 1\right)} - \frac{3 \cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} - 1\right)} \]derivative algebra simplifyDifferentiate the inner linear function 5*x. Multiply the constants. Simplify the fractions.✓ Proved
- \[ = \frac{3 \left(\frac{1}{\sin{\left(5 x \right)} + 1} - \frac{1}{\sin{\left(5 x \right)} - 1}\right) \cos{\left(5 x \right)}}{2} \]algebraFactor out the common term 3/2 * cos(5*x).✓ Proved
- \[ = - \frac{3 \cos{\left(5 x \right)}}{\left(\sin{\left(5 x \right)} - 1\right) \left(\sin{\left(5 x \right)} + 1\right)} \]algebraFind a common denominator for the terms in the parentheses.✓ Proved
- \[ = - \frac{3 \cos{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)} - 1} \]algebra algebraSimplify the numerator and denominator. Multiply the terms and simplify.✓ Proved
- \[ = \frac{3 \cos{\left(5 x \right)}}{1 - \sin^{2}{\left(5 x \right)}} \]algebraDistribute the negative sign into the denominator.✓ Proved
- \[ = \frac{3}{\cos{\left(5 x \right)}} \]rewrite simplifyUse the trigonometric identity 1 - sin^2(u) = cos^2(u). Cancel the common cos(5*x) term.✓ Proved
- \[ = 3 \sec{\left(5 x \right)} \]rewriteRewrite 1/cos(u) as sec(u).✓ Proved
Answer \( \frac{3}{\cos{\left(5 x \right)}} \)
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 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 4 | ✓ 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 |
| 5 | ✓ 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 |
| 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 undefined where sin(5*x)**2 - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 undefined where 1 - sin(5*x)**2 = 0 |
| 13 | ✓ 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 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and trigonometric identities are applied correctly and labeled appropriately.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and trigonometric identities are applied correctly and labeled appropriately.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 changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The final answer matches the stated answer.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-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.