Derivative of \( \displaystyle \frac{\left(3 \sin{\left(2 x + 1 \right)} - 3 \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \)
Problem 2.24 · hard
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \).
- \[ \frac{d}{d x} \frac{\left(3 \sin{\left(2 x + 1 \right)} - 3 \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]constant-multipleExtract the constant coefficient.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]productApply the product rule to the remaining function.✓ Proved
- \[ = \frac{3 \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) \frac{d}{d x} e^{2 x + 1}}{4} + \frac{3 e^{2 x + 1} \frac{d}{d x} \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right)}{4} \]chainApply the chain rule to the terms in the product.✓ Proved
- \[ = \frac{3 \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{2} + \frac{3 \left(2 \sin{\left(2 x + 1 \right)} + 2 \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]trigCompute the derivatives of the sine and cosine terms.✓ Proved
- \[ = 3 e^{2 x + 1} \sin{\left(2 x + 1 \right)} \]algebra simplify simplifyDistribute the exponential term. Combine like terms. Multiply the remaining constant and simplify.✓ Proved
Answer \( 3 e^{2 x + 1} \sin{\left(2 x + 1 \right)} \)
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (style) — Step 3 is labeled "chain" but it actually applies the product rule to the outer product. The chain rule is only used inside the derivative of sin(2*x+1)-cos(2*x+1), which is not explicitly separated. This mislabeling violates the rule‑granularity requirement.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product), which was already initiated in Step 2. The actual application of the chain rule occurs in Step 4 when differentiating the inner functions sin(2x+1) and exp(2x+1). Furthermore, Step 3 applies the chain rule to the inner derivatives implicitly within the same step as expanding the product structure, violating the one-rule-per-step constraint by conflating product expansion with inner differentiation.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product), which was already initiated in Step 2. The actual application of the chain rule occurs in Step 4 when differentiating the inner functions sin(2x+1) and exp(2x+1). Furthermore, Step 3 applies the chain rule to the inner derivatives implicitly within the same step as expanding the product structure, violating the one-rule-per-step constraint by conflating product expansion with inner differentiation.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 is labeled "chain" but it actually applies the product rule to the outer product. The chain rule is only used inside the derivative of sin(2*x+1)-cos(2*x+1), which is not explicitly separated. This mislabeling violates the rule‑granularity requirement.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 applies the chain rule to compute the derivatives of the factors, but the label 'chain' does not account for the differentiation of the exponential term or the subtraction inside the sine/cosine arguments which requires the 'sum' and 'derivative' rules as well. More critically, Step 4 is labeled 'trig' but actually performs the differentiation of the composite functions (applying chain rule results) and the derivative of the exponential, which should be labeled 'derivative' or 'chain'. The label 'trig' is not in the allowed vocabulary for differentiation steps (it usually implies trigonometric identities). Specifically, Step 3 claims to apply 'chain' but the transition from Step 2 to 3 is just writing out the product rule formula; the actual differentiation happens in Step 4. Step 3's label is misleading because it doesn't perform the differentiation, it just sets up the product rule structure. However, the prompt says 'A step that applies one rule to one term... is fine'. Step 3 applies the product rule structure? No, Step 2 was product. Step 3 is just expanding the product rule? No, Step 2 says 'Apply product rule' but leaves the derivatives as `Derivative(...)`. Step 3 computes those derivatives. So Step 3 is doing the differentiation. It uses chain rule for sin/cos and exp. But it also handles the subtraction. The label 'chain' is insufficient for the whole step. Also, Step 4 is labeled 'trig' but does nothing trigonometric, it just writes the results of the derivatives computed in step 3? No, Step 3 has `Derivative(...)` and Step 4 has the results. So Step 3 is the differentiation step. Step 3 label 'chain' is okay if we consider chain rule the dominant feature, but it ignores the linearity (sum/difference) and the exponential derivative. Step 4 label 'trig' is wrong because no trig identity is used, it's just the result of differentiation. Actually, looking closely: Step 3 goes from `Derivative(u*v)` to `u'*v + u*v'`. This is the Product Rule. Step 2 was labeled 'product' but didn't expand it. Step 3 expands it. So Step 3 should be 'product'. Step 4 computes the specific derivatives. Step 4 label 'trig' is wrong; it should be 'derivative' or 'chain'. The label 'trig' is not a valid differentiation rule in the list for computing derivatives of sin/cos/exp. The list has 'derivative', 'chain', 'trig'. 'Trig' usually means identities. Computing d/dx sin(x) is 'derivative' or 'chain'. Calling it 'trig' is a label defect.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the product rule as "chain" and step 4 incorrectly labels the chain rule as "trig". Each step should use a single, correctly named rule from the allowed vocabulary.qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product terms). The chain rule is applied in Step 4 to compute the specific derivatives of the composite functions, so Step 3 should be labeled 'product'.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 3 applies both the product rule and the chain rule in one line but labels only "chain". Each step must change only one thing and name the correct rule; this step violates both requirements.qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 applies the chain rule to compute the derivatives of the factors, but the label 'chain' does not account for the differentiation of the exponential term or the trigonometric terms themselves, which requires 'derivative' or 'trig' rules. More critically, Step 3 combines the application of the product rule (from Step 2) with the evaluation of the derivatives inside it, violating the 'one thing per step' constraint by performing the differentiation of the components within the same step as the structural expansion. Step 4 then claims to compute derivatives using 'trig', but the derivatives were already computed in Step 3. The labeling is inconsistent and the granularity of Step 3 is defective.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 3 incorrectly labels the application of the chain rule, which actually occurs in step 4. This misleads students about when the chain rule is applied.gpt-oss:20b: fail (error) 2026-09-19 — Step 3 applies both the product rule and the chain rule at once, yet it is labeled only as "chain". Each step must change only one thing and use the correct rule name. This violates the contract.gpt-oss:20b: fail 2026-09-17 — Step 3 incorrectly labels the rule as "chain" when it is actually the product rule, and step 4 labels it as "trig" while it applies the chain rule to the inner sine and cosine functions. This mislabeling could mislead a student.deepseek-r1:70b: fail 2026-09-17 — Step 3 mislabels the product rule as the chain rule, which could mislead students.
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.