Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(4 x \right)} - 1 \right)}}{8} + \frac{\ln{\left(\sin{\left(4 x \right)} + 1 \right)}}{8} \)
Problem 2.159 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(4 x \right)} - 1 \right)}}{8} + \frac{\ln{\left(\sin{\left(4 x \right)} + 1 \right)}}{8} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(4 x \right)} - 1 \right)}}{8} + \frac{\ln{\left(\sin{\left(4 x \right)} + 1 \right)}}{8}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(4 x \right)} - 1 \right)}}{8}\right) + \frac{d}{d x} \frac{\ln{\left(\sin{\left(4 x \right)} + 1 \right)}}{8} \]sum constant-multipleApply the sum rule. Factor out the constants.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(4 x \right)} - 1 \right)}}{8} + \frac{\frac{d}{d x} \ln{\left(\sin{\left(4 x \right)} + 1 \right)}}{8} \]constant-multipleApply the constant multiple rule to each term.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(4 x \right)} + 1\right)}{8 \left(\sin{\left(4 x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\sin{\left(4 x \right)} - 1\right)}{8 \left(\sin{\left(4 x \right)} - 1\right)} \]chainApply the chain rule to the logarithmic functions.✓ Proved
- \[ = \frac{\cos{\left(4 x \right)} \frac{d}{d x} 4 x}{8 \left(\sin{\left(4 x \right)} + 1\right)} - \frac{\cos{\left(4 x \right)} \frac{d}{d x} 4 x}{8 \left(\sin{\left(4 x \right)} - 1\right)} \]chainApply the chain rule to the sine functions.✓ Proved
- \[ = \frac{\cos{\left(4 x \right)}}{2 \left(\sin{\left(4 x \right)} + 1\right)} - \frac{\cos{\left(4 x \right)}}{2 \left(\sin{\left(4 x \right)} - 1\right)} \]derivative algebra simplify algebraDifferentiate the innermost function 4*x. Simplify the coefficients. Reduce the fractions. Factor out the common term 1/2 and cos(4*x).✓ Proved
- \[ = \frac{\frac{\left(\sin{\left(4 x \right)} - 1\right) \cos{\left(4 x \right)}}{2} - \frac{\left(\sin{\left(4 x \right)} + 1\right) \cos{\left(4 x \right)}}{2}}{\left(\sin{\left(4 x \right)} - 1\right) \left(\sin{\left(4 x \right)} + 1\right)} \]algebraFind a common denominator.✓ Proved
- \[ = - \frac{\cos{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)} - 1} \]algebra algebra simplify simplifyExpand the products in the numerator and denominator. Distribute the negative sign. Combine like terms in the numerator. Simplify the fraction.✓ Proved
- \[ = \frac{\cos{\left(4 x \right)}}{1 - \sin^{2}{\left(4 x \right)}} \]algebraMultiply the numerator and denominator by -1.✓ Proved
- \[ = \frac{1}{\cos{\left(4 x \right)}} \]rewrite simplifyUse the identity 1 - sin(u)^2 = cos(u)^2. Cancel the common factor of cos(4*x).✓ Proved
- \[ = \sec{\left(4 x \right)} \]rewriteRewrite 1/cos(u) as sec(u).✓ Proved
Answer \( - \frac{\cos{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)} - 1} \)
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(4*x) + 1 = 0 undefined where sin(4*x) - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) + 1 = 0 undefined where sin(4*x) - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) + 1 = 0 undefined where sin(4*x) - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) + 1 = 0 undefined where sin(4*x) - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) + 1 = 0 undefined where sin(4*x) - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) + 1 = 0 undefined where sin(4*x) - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) + 1 = 0 undefined where sin(4*x) - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) + 1 = 0 undefined where sin(4*x) - 1 = 0 undefined where sin(4*x)**2 - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x)**2 - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x)**2 - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x)**2 - 1 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x)**2 - 1 = 0 undefined where 1 - sin(4*x)**2 = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - sin(4*x)**2 = 0 undefined where cos(4*x) = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) = 0 |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*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 sin(4*x)**2 - 1 = 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 and algebraic simplifications. Each step isolates a single transformation, 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 isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules with proper granularity, labeling each step with a valid rule from the fixed vocabulary. The algebraic simplifications and trigonometric rewrites are accurate and clearly justified.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 changes only one aspect of the expression, and the labels accurately reflect the operations performed.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. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 10’s note claims that both 1/2 and cos(4*x) are factored out, but the expression only factors 1/2; the note misleads the student about the algebraic manipulation performed.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.