Derivative of \( \displaystyle \frac{5 \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{5 \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \)
Problem 2.431 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{5 \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \).
- \[ \frac{d}{d x} \left(\frac{5 \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{5 \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8}\right) \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{5 \frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \]algebra logarithmicDistribute the constant factor. Apply the chain rule for the logarithm.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \left(\cos{\left(4 x \right)} + 1\right)}{8 \left(\cos{\left(4 x \right)} + 1\right)} + \frac{5 \frac{d}{d x} \left(\cos{\left(4 x \right)} - 1\right)}{8 \left(\cos{\left(4 x \right)} - 1\right)} \]trigDifferentiate the inner cosine functions.✓ Proved
- \[ = \frac{5 \sin{\left(4 x \right)} \frac{d}{d x} 4 x}{8 \left(\cos{\left(4 x \right)} + 1\right)} - \frac{5 \sin{\left(4 x \right)} \frac{d}{d x} 4 x}{8 \left(\cos{\left(4 x \right)} - 1\right)} \]chainApply the chain rule to the argument 4*x.✓ Proved
- \[ = \frac{5 \sin{\left(4 x \right)}}{2 \left(\cos{\left(4 x \right)} + 1\right)} - \frac{5 \sin{\left(4 x \right)}}{2 \left(\cos{\left(4 x \right)} - 1\right)} \]algebra constant-multipleEvaluate the derivative of 4*x. Simplify the coefficients.✓ Proved
- \[ = \frac{5 \left(\frac{1}{\cos{\left(4 x \right)} + 1} - \frac{1}{\cos{\left(4 x \right)} - 1}\right) \sin{\left(4 x \right)}}{2} \]algebraFactor out 4*sin(4*x) and simplify 4 * 5/8.✓ Proved
- \[ = - \frac{5 \sin{\left(4 x \right)}}{\left(\cos{\left(4 x \right)} - 1\right) \left(\cos{\left(4 x \right)} + 1\right)} \]algebra algebraFind a common denominator for the terms inside the parentheses. Simplify the numerator in the parentheses.✓ Proved
- \[ = - \frac{5 \sin{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)} - 1} \]algebraExpand the denominator using the difference of squares.✓ Proved
- \[ = \frac{5}{\sin{\left(4 x \right)}} \]trig algebra simplifyUse the identity cos(u)**2 - 1 = -sin(u)**2. Simplify the signs and the fraction. Final simplification.✓ Proved
Answer \( \frac{5}{\sin{\left(4 x \right)}} \)
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.
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 undefined where cos(4*x) - 1 = 0 |
| 5 | ✓ 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 |
| 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 undefined where cos(4*x)**2 - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x)**2 - 1 = 0 undefined where sin(4*x) = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) = 0 |
| 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: fail (style) — Step 3 incorrectly labels an algebraic regrouping as "logarithmic", and step 4 incorrectly labels the application of the logarithm derivative rule as "trig". These mislabelings violate the rule‑granularity requirement.qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'logarithmic' but performs no differentiation; it merely factors out a constant, which should be labeled 'algebra'. Step 4 is labeled 'trig' but applies the chain rule for the logarithm (derivative of log(u) is u'/u), which should be labeled 'chain' or 'derivative'. The labels do not match the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is labeled 'logarithmic' but performs no differentiation; it merely factors out a constant, which should be labeled 'algebra'. Step 4 is labeled 'trig' but applies the chain rule for the logarithm (derivative of log(u) is u'/u), which should be labeled 'chain' or 'derivative'. The labels do not match the operations performed.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels an algebraic regrouping as "logarithmic", and step 4 incorrectly labels the application of the logarithm derivative rule as "trig". These mislabelings violate the rule‑granularity requirement.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 is labeled 'logarithmic' but the step merely factors out a constant (algebraic manipulation); the actual differentiation of the logarithm occurs in Step 4. Step 4 is labeled 'trig' but performs the differentiation of the log and the inner function, which should be labeled 'chain' or 'derivative'.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 is labeled 'logarithmic' but performs no differentiation; it merely factors out the constant 5/8, which should be labeled 'algebra' or 'constant-multiple'. Step 4 is labeled 'trig' but applies the derivative of the logarithm (chain rule), so it should be labeled 'logarithmic' or 'chain'. The labels are swapped and incorrect for the operations performed.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the algebraic combination of two derivative terms as "logarithmic"; it should be "algebra". Step 4 applies the logarithmic derivative rule but labels the step as "trig"; it should be "logarithmic". These mislabelings violate the rule‑granularity requirement.
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.