Derivative of \( \displaystyle \frac{\cos{\left(4 x + 2 \right)}}{4 x + 2} \)
Problem 2.313 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(4 x + 2 \right)}}{4 x + 2} \).
- \[ \frac{d}{d x} \frac{\cos{\left(4 x + 2 \right)}}{4 x + 2} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \cos{\left(4 x + 2 \right)}}{4 x + 2} - \frac{\cos{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right)}{\left(4 x + 2\right)^{2}} \]quotientApply the quotient rule.✓ Proved
- \[ = - \frac{\sin{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right)}{4 x + 2} - \frac{\cos{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right)}{\left(4 x + 2\right)^{2}} \]derivativeDifferentiate the numerator.✓ Proved
- \[ = - \frac{4 \sin{\left(4 x + 2 \right)}}{4 x + 2} - \frac{4 \cos{\left(4 x + 2 \right)}}{\left(4 x + 2\right)^{2}} \]derivative constant-multipleDifferentiate the denominator. Factor out the constant 4.✓ Proved
- \[ = \frac{- 4 \left(4 x + 2\right) \sin{\left(4 x + 2 \right)} - 4 \cos{\left(4 x + 2 \right)}}{\left(4 x + 2\right)^{2}} \]algebra algebraCombine the terms over a common denominator. Factor out -4 from the numerator.✓ Proved
Answer \( \frac{\left(- 4 x - 2\right) \sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}}{\left(2 x + 1\right)^{2}} \)
✓ 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 4*x + 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 2*x + 1 = 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) — The final simplification in step 7 incorrectly drops the factor of 4 in the denominator. The correct derivative is \displaystyle \frac{(-4x-2)\sin(4x+2)-\cos(4x+2)}{(2x+1)^2}, not \displaystyle -4\frac{\sin(4x+2)(4x+2)+\cos(4x+2)}{(4x+2)^2}.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 3 applies the chain rule to the numerator but leaves the derivative of the denominator in Step 4, violating the 'one rule per step' constraint by splitting the quotient rule application across two steps. Additionally, Step 3 is labeled 'derivative' but performs a chain rule application (differentiating a composite function), which should be labeled 'chain' or 'derivative' if interpreted as unfolding d/dx(sin), but the primary issue is the structural split of the quotient rule.
Senior review claude-sonnet-5-5, 2026-10-07: pass — The quotient rule is applied correctly and every line is SymPy-proved equal to the previous one. The final line matches the stated answer, and the steps and labels follow the contract.
gpt-oss:20b: dismiss — Line 7 is correct: (4x+2)^2 = 4(2x+1)^2, so -4(u sin u + cos u)/(4x+2)^2 equals the stated answer, and SymPy proved the final line equivalent to it.qwen3.6:27b-mlx: dismiss — Splitting the work into 'differentiate the numerator' and then evaluate the remaining inner derivative of 4x+2 is one change per step. The 'derivative' label is in the vocabulary and fits both steps, so no rule is misnamed.
Every verdict on record (13)
claude-sonnet-5-5: pass 2026-10-07 — The quotient rule is applied correctly and every line is SymPy-proved equal to the previous one. The final line matches the stated answer, and the steps and labels follow the contract.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies the chain rule to the numerator but leaves the derivative of the denominator in Step 4, violating the 'one rule per step' constraint by splitting the quotient rule application across two steps. Additionally, Step 3 is labeled 'derivative' but performs a chain rule application (differentiating a composite function), which should be labeled 'chain' or 'derivative' if interpreted as unfolding d/dx(sin), but the primary issue is the structural split of the quotient rule.gpt-oss:20b: fail (error) 2026-09-20 — The final simplification in step 7 incorrectly drops the factor of 4 in the denominator. The correct derivative is \displaystyle \frac{(-4x-2)\sin(4x+2)-\cos(4x+2)}{(2x+1)^2}, not \displaystyle -4\frac{\sin(4x+2)(4x+2)+\cos(4x+2)}{(4x+2)^2}.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'derivative' but applies the chain rule to differentiate the composite function cos(4x+2). The label 'derivative' is reserved for basic derivatives (like d/dx sin(x)), whereas this step requires the 'chain' rule. Additionally, Step 3 only differentiates the numerator while leaving the denominator's derivative unevaluated, which is a partial application not covered by the 'derivative' label alone in this context, but the primary defect is the mislabeling of the chain rule application.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 uses an unlabeled rule; every step must name a rule from the fixed vocabulary (e.g., "derivative").qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 applies the chain rule to the numerator but fails to label it as such, incorrectly labeling it 'derivative' while leaving the denominator's derivative unevaluated. Step 4 then evaluates the denominator's derivative, violating the 'one rule per step' constraint by splitting the application of the quotient rule's components across two steps with incorrect labels.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 applies the chain rule to differentiate the numerator but fails to label it as such; the label 'derivative' is reserved for basic derivatives (like d/dx sin(x)), whereas this step requires the 'chain' rule for the composite function sin(4x+2). Additionally, Step 3 only differentiates the numerator while leaving the denominator's derivative unevaluated, which is a partial application of the quotient rule expansion, but the primary defect is the mislabeling of the chain rule application.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 4’s note says "Differentiate the denominator," but the step actually substitutes the derivative of the inner function 4*x+2 in both terms. This misleads a student about what operation is being performed.deepseek-r1:70b: fail 2026-09-17 — Step 7's note about factoring out -4 is misleading as the stated answer doesn't reflect this factor.
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.