Home›Calculus 1›Quotient rule›Problem 2.521

Derivative of \( \displaystyle \frac{\cos{\left(4 x \right)}}{4 x} \)

Problem 2.521 · hard Video

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 \( - \frac{\sin{\left(4 x \right)}}{x} - \frac{\cos{\left(4 x \right)}}{4 x^{2}} \)

Top: the graph of f = cos(4x)/(4x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -sin(4x)/x - cos(4x)/(4x²).

✓ 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. How we verify

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: pass
qwen3.6:27b-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.

Senior review claude-sonnet-5-5, 2026-10-09: pass — Every line is equivalent to the previous one and the rule labels fit. The note on step 9 mentions only the first term, but the step's simplification of both terms is a single simplify step and its result is correct.

qwen3.6:27b-mlx: dismiss — Line 9 correctly reduces both terms (4/(4x) = 1/x and 4/(4x)^2 = 1/(4x^2)), so no stray factor of 4 remains and the result matches the stated answer; 'simplify' is a legitimate label, and SymPy proved the step.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-09 — Every line is equivalent to the previous one and the rule labels fit. The note on step 9 mentions only the first term, but the step's simplification of both terms is a single simplify step and its result is correct.
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 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-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.