Derivative of \( \displaystyle \frac{\sin{\left(4 x + 1 \right)}}{4 x + 2} \)
Problem 2.330 · hard
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(4 x + 1 \right)}}{4 x + 2} \).
- \[ \frac{d}{d x} \frac{\sin{\left(4 x + 1 \right)}}{4 x + 2} \]derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
- \[ = \sin{\left(4 x + 1 \right)} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{\frac{d}{d x} \sin{\left(4 x + 1 \right)}}{4 x + 2} \]productApply the product rule.✓ Proved
- \[ = \sin{\left(4 x + 1 \right)} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{4 \cos{\left(4 x + 1 \right)}}{4 x + 2} \]chainApply the chain rule to the first term.✓ Proved
- \[ = \frac{4 \cos{\left(4 x + 1 \right)}}{4 x + 2} - \frac{\sin{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 2\right)}{\left(4 x + 2\right)^{2}} \]powerApply the power rule to the second term.✓ Proved
- \[ = \frac{4 \cos{\left(4 x + 1 \right)}}{4 x + 2} - \frac{4 \sin{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]chain algebra algebraApply the chain rule to the inner function. Simplify the expression by combining constants. Rewrite with negative exponents as fractions.✓ Proved
- \[ = \frac{4 \left(4 x + 2\right) \cos{\left(4 x + 1 \right)} - 4 \sin{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]algebraFind a common denominator.✓ Proved
- \[ = \frac{16 x \cos{\left(4 x + 1 \right)} - 4 \sin{\left(4 x + 1 \right)} + 8 \cos{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]algebraDistribute the term in the numerator.✓ Proved
- \[ = \frac{\left(16 x + 8\right) \cos{\left(4 x + 1 \right)} - 4 \sin{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]simplifyFactor out the common constant 4.✓ Proved
Answer \( \frac{\left(4 x + 2\right) \cos{\left(4 x + 1 \right)} - \sin{\left(4 x + 1 \right)}}{\left(2 x + 1\right)^{2}} \)
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 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 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| 11 | ✓ 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: passdeepseek-r1:70b: fail (error) — Step 5 applies both power and chain rules, which is a defect as each step should only apply one rule.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final result is algebraically equivalent to the stated answer.
Every verdict on record (13)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 10 applies algebraic distribution but the label 'algebra' is acceptable; however, Step 11 factors out a 4 from the numerator but leaves the denominator as (4*x + 2)**2, resulting in a form that does not match the stated answer which simplifies the denominator to (2*x + 1)**2. The solution stops short of the final simplification required to match the stated answer, or rather, the stated answer is a further simplification of step 11 that is not shown. More critically, Step 6 is labeled 'chain' but performs the differentiation of the inner function (derivative of 4x+2 is 4), which is technically a 'derivative' step applied to the inner function, while the outer power rule was applied in Step 5. The combination of power rule and chain rule is often split, but here Step 5 applied the power rule to the outer function leaving the inner derivative, and Step 6 computes that inner derivative. This is acceptable granularity. The main issue is that the final step shown (11) does not match the stated answer. The stated answer has denominator (2*x+1)^2, which is (4*x+2)^2 / 4. The numerator in step 11 has a factor of 4. So step 11 is 4 * [ (4*x+2)cos(...) - sin(...) ] / (4*x+2)^2. The stated answer is [ (4*x+2)cos(...) - sin(...) ] / (2*x+1)^2. Since (4*x+2)^2 = 4*(2*x+1)^2, the 4 in the numerator cancels with the 4 from the denominator expansion. The solution fails to show this final simplification step to reach the stated answer. However, the prompt says 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. It does not say the last line equals the stated answer. But usually, the last line should be the answer. Let's look closer. Step 11 is not equal to the stated answer? Step 11: 4*(...) / (4x+2)^2. Stated: (...) / (2x+1)^2. (4x+2)^2 = 16x^2 + 16x + 4 = 4(4x^2 + 4x + 1) = 4(2x+1)^2. So Step 11 = 4*(...) / 4*(2x+1)^2 = (...) / (2x+1)^2. They are equal. So the algebra is fine. Is there a rule violation? Step 6: 'Apply the chain rule to the inner function'. The rule 'chain' is used. Step 5 used 'power'. This is a valid split. Step 4: 'Apply the chain rule to the first term'. The first term was Derivative(sin(4x+1), x). The derivative of sin(u) is cos(u)*u'. Step 4 produces 4*cos(4x+1)*(4x+2)^(-1). This applies the chain rule to sin(4x+1). This is correct. Step 3 applies product rule. Correct. Step 2 rewrites quotient as product. Correct. Step 1 is just stating the problem. Correct. Wait, look at Step 6 again. The note says 'Apply the chain rule to the inner function'. The step computes Derivative(4*x + 2, x) -> 4. This is a 'derivative' step (derivative of a linear function). The label 'chain' is incorrect because the chain rule was already invoked in Step 5 (power rule with chain). Step 5 applied the power rule to (4x+2)^(-1), resulting in -1*(4x+2)^(-2) * Derivative(4x+2, x). Step 6 evaluates that derivative. Evaluating the derivative of a polynomial is a 'derivative' step, not 'chain'. The chain rule is the rule that says d/dx f(g(x)) = f'(g(x))g'(x). Step 5 applied the 'power' part of the chain rule (or rather, the power rule which is a specific case). Step 6 just computes the derivative of the inside. Calling Step 6 'chain' is misleading/wrong label. It should be 'derivative'. Similarly, Step 4 computes Derivative(sin(4x+1), x) -> 4cos(4x+1). This is applying the chain rule to the sine function. So Step 4 is correctly labeled 'chain'. Step 6 is incorrectly labeled 'chain' when it is just differentiating the inner linear term. Actually, one could argue Step 5+6 together are the chain rule application for the power function. But they are split. Step 5 is 'power'. Step 6 is 'derivative'. Labeling Step 6 as 'chain' is a defect.deepseek-r1:70b: fail (error) 2026-09-19 — Step 5 applies both power and chain rules, which is a defect as each step should only apply one rule.gpt-oss:20b: fail (error) 2026-09-19 — The final expression has a denominator of (4*x+2)^2, which equals 4*(2*x+1)^2. The expected result has denominator (2*x+1)^2, so a factor of 4 is missing. The solution incorrectly omitted this factor.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final answer is algebraically equivalent to the stated answer.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the transformations performed, and the final result is algebraically equivalent to the stated answer.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (error) 2026-09-18 — Step 11 factors a 4 out of the numerator but leaves the denominator unchanged, producing an expression that is not algebraically equivalent to the previous step. The correct simplification would require dividing the denominator by 4 as well, or keeping the factor inside the numerator. This makes the final result incorrect.
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.