Derivative of \( \displaystyle \frac{3 \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{3 \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \)
Problem 2.1337 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{3 \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \).
- \[ \frac{d}{d x} \left(\frac{3 \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{3 \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \]constant-multiple algebraApply the constant multiple rule to each term. Factor out the common constant 3/8.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{3 \frac{d}{d x} \left(\cos{\left(4 x \right)} + 1\right)}{8 \left(\cos{\left(4 x \right)} + 1\right)} \]algebraDistribute the subtraction.✓ Proved
- \[ = - \frac{3 \frac{d}{d x} \left(\cos{\left(4 x \right)} + 1\right)}{8 \left(\cos{\left(4 x \right)} + 1\right)} + \frac{3 \frac{d}{d x} \left(\cos{\left(4 x \right)} - 1\right)}{8 \left(\cos{\left(4 x \right)} - 1\right)} \]logarithmicApply the derivative rule for the natural logarithm.✓ Proved
- \[ = - \frac{3 \frac{d}{d x} \cos{\left(4 x \right)}}{8 \left(\cos{\left(4 x \right)} + 1\right)} + \frac{3 \frac{d}{d x} \cos{\left(4 x \right)}}{8 \left(\cos{\left(4 x \right)} - 1\right)} \]constantDifferentiate the constant term -1 and 1.✓ Proved
- \[ = \frac{3 \sin{\left(4 x \right)} \frac{d}{d x} 4 x}{8 \left(\cos{\left(4 x \right)} + 1\right)} - \frac{3 \sin{\left(4 x \right)} \frac{d}{d x} 4 x}{8 \left(\cos{\left(4 x \right)} - 1\right)} \]trigApply the derivative rule for the cosine function.✓ Proved
- \[ = \frac{3 \sin{\left(4 x \right)}}{2 \left(\cos{\left(4 x \right)} + 1\right)} - \frac{3 \sin{\left(4 x \right)}}{2 \left(\cos{\left(4 x \right)} - 1\right)} \]chain algebra algebraApply the chain rule to the inner function 4*x. Simplify the products within the parentheses. Factor out 4 and multiply with 3/8.✓ Proved
- \[ = \frac{3 \left(\frac{1}{\cos{\left(4 x \right)} + 1} - \frac{1}{\cos{\left(4 x \right)} - 1}\right) \sin{\left(4 x \right)}}{2} \]algebraFactor out sin(4*x).✓ Proved
- \[ = - \frac{3 \sin{\left(4 x \right)}}{\left(\cos{\left(4 x \right)} - 1\right) \left(\cos{\left(4 x \right)} + 1\right)} \]algebraCombine the fractions using a common denominator.✓ Proved
- \[ = - \frac{3 \sin{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)} - 1} \]algebraSimplify the numerator and the denominator.✓ Proved
- \[ = \frac{3}{\sin{\left(4 x \right)}} \]algebra algebra algebra simplifyUse the trigonometric identity cos(u)**2 - 1 = -sin(u)**2. Simplify the signs and the fraction. Multiply the terms. Cancel one sin(4*x) from the numerator and denominator.✓ Proved
Answer \( \frac{3}{\sin{\left(4 x \right)}} \)
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. A reviewer's objection to the explanation stands; see 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 |
| 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 undefined where sin(4*x) = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x) = 0 |
| 17 | ✓ 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 (error) — Step 4 applies two rules at once: it both distributes the subtraction and applies the logarithmic derivative rule to the second term. Each step must change only one thing, so this step is invalid.qwen3.6:27b-mlx: fail (error) — Step 4 is labeled 'algebra' but performs differentiation of the second term using the chain rule (introducing the derivative of the inner function). This violates the single-rule constraint and mislabels a differentiation step as algebraic manipulation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 4 is labeled 'algebra' but performs differentiation of the second term using the chain rule (introducing the derivative of the inner function). This violates the single-rule constraint and mislabels a differentiation step as algebraic manipulation.gpt-oss:20b: fail (error) 2026-10-03 — Step 4 applies two rules at once: it both distributes the subtraction and applies the logarithmic derivative rule to the second term. Each step must change only one thing, so this step is invalid.qwen3.6:27b-mlx: fail (style) 2026-09-30 — Step 4 is labeled 'algebra' but performs a differentiation (applying the chain rule to the second log term), which violates the one-rule-per-step constraint and mislabels the operation. Step 8 is labeled 'chain' but only evaluates the derivative of the inner linear function (which is a 'derivative' or 'constant' step), having already applied the chain rule structure in previous steps.gpt-oss:20b: fail (error) 2026-09-30 — Step 4 applies two rules at once: it replaces the derivative of the second logarithm with the logarithmic derivative rule while also distributing the subtraction. The label "algebra" is incorrect for the logarithmic differentiation applied here.
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.