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 2.1438 · 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
- \[ = \frac{d}{d x} \frac{\ln{\left(\sin{\left(4 x \right)} - 1 \right)}}{8} - \frac{d}{d x} \frac{\ln{\left(\sin{\left(4 x \right)} + 1 \right)}}{8} \]sumApply the sum rule.✓ 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-multipleFactor out the constant 1/8.✓ 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 terms.✓ Proved
- \[ = - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \sin{\left(4 x \right)}}{8 \left(\sin{\left(4 x \right)} + 1\right)} + \frac{- \frac{d}{d x} 1 + \frac{d}{d x} \sin{\left(4 x \right)}}{8 \left(\sin{\left(4 x \right)} - 1\right)} \]sumDifferentiate the terms inside the parentheses.✓ 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 simplifyCompute the derivative of sin(4*x) and the constant 1. Simplify the expression by removing zero.✓ Proved
- \[ = - \frac{4 \cos{\left(4 x \right)}}{8 \sin{\left(4 x \right)} + 8} + \frac{4 \cos{\left(4 x \right)}}{8 \sin{\left(4 x \right)} - 8} \]algebraMultiply the terms out.✓ Proved
- \[ = - \frac{\cos{\left(4 x \right)}}{2 \sin{\left(4 x \right)} + 2} + \frac{\cos{\left(4 x \right)}}{2 \sin{\left(4 x \right)} - 2} \]simplifySimplify the fractions by dividing by 4.✓ Proved
- \[ = \frac{- \left(\sin{\left(4 x \right)} - 1\right) \cos{\left(4 x \right)} + \left(\sin{\left(4 x \right)} + 1\right) \cos{\left(4 x \right)}}{\left(\sin{\left(4 x \right)} + 1\right) \left(2 \sin{\left(4 x \right)} - 2\right)} \]algebraFind a common denominator.✓ Proved
- \[ = \frac{2 \cos{\left(4 x \right)}}{2 \sin^{2}{\left(4 x \right)} - 2} \]algebra simplifyDistribute the cosine term in the numerator. Combine like terms in the numerator.✓ Proved
- \[ = \frac{\cos{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)} - 1} \]simplifyCancel the common factor of 2.✓ Proved
- \[ = - \frac{\cos{\left(4 x \right)}}{1 - \sin^{2}{\left(4 x \right)}} \]algebraRewrite the denominator using a sign change.✓ Proved
- \[ = - \frac{1}{\cos{\left(4 x \right)}} \]rewrite simplifyUse the Pythagorean identity 1 - sin^2(u) = cos^2(u). Cancel one factor of cos(4*x).✓ Proved
- \[ = - \sec{\left(4 x \right)} \]rewriteUse the identity 1/cos(u) = 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 undefined where sin(4*x) + 1 = 0 undefined where sin(4*x) - 1 = 0 |
| 5 | ✓ 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 |
| 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 undefined where 8*sin(4*x) - 8 = 0 undefined where 8*sin(4*x) + 8 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 8*sin(4*x) - 8 = 0 undefined where 8*sin(4*x) + 8 = 0 undefined where 2*sin(4*x) + 2 = 0 undefined where 2*sin(4*x) - 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*sin(4*x) + 2 = 0 undefined where 2*sin(4*x) - 2 = 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 2*sin(4*x) - 2 = 0 undefined where 2*sin(4*x)**2 - 2 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*sin(4*x)**2 - 2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*sin(4*x)**2 - 2 = 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 undefined where 1 - sin(4*x)**2 = 0 |
| 15 | ✓ 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 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) = 0 |
| 17 | ✓ 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: passqwen3.6:27b-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 (4)
qwen3.6:27b-mlx: pass 2026-10-03 — 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 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.