Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{4} - \frac{\ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{4} \)
Problem 2.84 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{4} - \frac{\ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{\ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{4} - \frac{\ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{4}\right) \]constantStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{4} - \frac{\frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{4} \]constant-multiple algebraDistribute the constant factor. Factor out the common constant.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{4} - \frac{\frac{d}{d x} \left(\cos{\left(4 x \right)} + 1\right)}{4 \left(\cos{\left(4 x \right)} + 1\right)} \]chainApply the chain rule to the second term.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\cos{\left(4 x \right)} + 1\right)}{4 \left(\cos{\left(4 x \right)} + 1\right)} + \frac{\frac{d}{d x} \left(\cos{\left(4 x \right)} - 1\right)}{4 \left(\cos{\left(4 x \right)} - 1\right)} \]chainApply the chain rule to the first term.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(4 x \right)}}{4 \left(\cos{\left(4 x \right)} + 1\right)} + \frac{\frac{d}{d x} \cos{\left(4 x \right)}}{4 \left(\cos{\left(4 x \right)} - 1\right)} \]derivativeDifferentiate the inner cosine function.✓ Proved
- \[ = \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)} + 1} - \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)} - 1} \]chain algebra algebra constant-multipleApply the chain rule to the argument 4*x. Simplify the products. Simplify the signs. Distribute the 1/4.✓ Proved
- \[ = \left(\frac{1}{\cos{\left(4 x \right)} + 1} - \frac{1}{\cos{\left(4 x \right)} - 1}\right) \sin{\left(4 x \right)} \]algebraFactor out sin(4*x).✓ Proved
- \[ = - \frac{2 \sin{\left(4 x \right)}}{\left(\cos{\left(4 x \right)} - 1\right) \left(\cos{\left(4 x \right)} + 1\right)} \]algebraFind a common denominator.✓ Proved
- \[ = - \frac{2 \sin{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)} - 1} \]algebra simplify algebraExpand the numerator. Combine like terms in the numerator. Multiply sin(4*x) into the fraction.✓ Proved
- \[ = \frac{2 \sin{\left(4 x \right)}}{1 - \cos^{2}{\left(4 x \right)}} \]algebraMultiply numerator and denominator by -1.✓ Proved
- \[ = \frac{2}{\sin{\left(4 x \right)}} \]trig simplifyUse the identity 1 - cos(u)^2 = sin(u)^2. Cancel one sin(4*x) term.✓ Proved
- \[ = 2 \csc{\left(4 x \right)} \]rewriteRewrite 1/sin(u) as csc(u).✓ Proved
Answer \( \frac{2}{\sin{\left(4 x \right)}} \)
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 undefined where cos(4*x) + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(4*x) + 1 = 0 undefined where cos(4*x) - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) + 1 = 0 undefined where cos(4*x) - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) + 1 = 0 undefined where cos(4*x) - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) + 1 = 0 undefined where cos(4*x) - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) + 1 = 0 undefined where cos(4*x) - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) + 1 = 0 undefined where cos(4*x) - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) + 1 = 0 undefined where cos(4*x) - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) + 1 = 0 undefined where cos(4*x) - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) + 1 = 0 undefined where cos(4*x) - 1 = 0 undefined where cos(4*x)**2 - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x)**2 - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x)**2 - 1 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x)**2 - 1 = 0 undefined where 1 - cos(4*x)**2 = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - cos(4*x)**2 = 0 undefined where sin(4*x) = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) = 0 |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) = 0 csc has poles at multiples of pi |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(4*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 and algebraic simplifications in single-step increments. All labels are appropriate for the operations performed.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate for 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 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 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate for the transformations 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 step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications and trigonometric identities are applied correctly.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 1 incorrectly labels the rule as 'constant'; the derivative of a sum/difference requires the linearity rule, not a constant rule.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.