Derivative of \( \displaystyle \frac{\cos{\left(4 x \right)}}{4 x} \)
Problem 2.521 · hard
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(4 x \right)}}{4 x} \).
- \[ \frac{d}{d x} \frac{\cos{\left(4 x \right)}}{4 x} \]derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
- \[ = \cos{\left(4 x \right)} \frac{d}{d x} \frac{1}{4 x} + \frac{\frac{d}{d x} \cos{\left(4 x \right)}}{4 x} \]productApply the product rule.✓ Proved
- \[ = \cos{\left(4 x \right)} \frac{d}{d x} \frac{1}{4 x} - \frac{\sin{\left(4 x \right)}}{x} \]chain constant-multipleApply the chain rule to the cosine term. Factor out the constant 4.✓ Proved
- \[ = - \frac{\sin{\left(4 x \right)}}{x} - \frac{\cos{\left(4 x \right)}}{4 x^{2}} \]chain algebra algebra simplifyApply the chain rule to the power term. Simplify the signs and distribute the terms. Rewrite the negative exponents as fractions. Cancel the common factor of 4 in the first term.✓ Proved
Answer \( - \frac{\sin{\left(4 x \right)}}{x} - \frac{\cos{\left(4 x \right)}}{4 x^{2}} \)
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 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 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: fail (error) — Step 9 claims to cancel the common factor of 4 in the first term, but the second term still retains a factor of 4 in the numerator (which should also be cancelled to match the stated answer). The step applies simplification to only one term while leaving the other unchanged, violating the 'one thing' rule or resulting in an incorrect final expression if the second term is not also simplified.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 9 claims to cancel the common factor of 4 in the first term, but the second term still retains a factor of 4 in the numerator (which should also be cancelled to match the stated answer). The step applies simplification to only one term while leaving the other unchanged, violating the 'one thing' rule or resulting in an incorrect final expression if the second term is not also simplified.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 9 applies simplification to both terms of the sum (canceling 4 in the first term and simplifying the denominator in the second), violating the one-change-per-step rule.gpt-oss:20b: pass 2026-09-20
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.