Derivative of \( \displaystyle \frac{\left(- \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{4 x}}{8} \)
Problem 2.188 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{4 x}}{8} \).
- \[ \frac{d}{d x} \frac{\left(- \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]constantPull out the constant factor 1/8.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- e^{4 x} \sin{\left(4 x \right)}\right)}{8} + \frac{\frac{d}{d x} e^{4 x} \cos{\left(4 x \right)}}{8} \]sumApply the sum rule.✓ Proved
- \[ = \frac{e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{8} + \frac{\cos{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} + \frac{\frac{d}{d x} \left(- e^{4 x} \sin{\left(4 x \right)}\right)}{8} \]productApply the product rule to the second term.✓ Proved
- \[ = - \frac{e^{4 x} \sin{\left(4 x \right)}}{2} + \frac{e^{4 x} \cos{\left(4 x \right)}}{2} + \frac{\frac{d}{d x} \left(- e^{4 x} \sin{\left(4 x \right)}\right)}{8} \]chainApply the chain rule to the trigonometric and exponential terms.✓ Proved
- \[ = - e^{4 x} \sin{\left(4 x \right)} \]product algebra simplify simplifyApply the product rule to the first term. Combine the terms into a single fraction. Combine like terms in the numerator. Simplify the final expression.✓ Proved
Answer \( - e^{4 x} \sin{\left(4 x \right)} \)
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 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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 (error) — Step 5 incorrectly applies the chain rule to the first term: it replaces the derivative of −sin(4x) with the derivative of cos(4x). The correct derivative of −sin(4x) is −cos(4x)×4, not −sin(4x)×4. This single step violates the rule‑granularity requirement and leads to an incorrect final result.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to differentiate cos(4*x) and exp(4*x) but fails to apply the product rule to the first term (-sin(4*x)*exp(4*x)), which is still wrapped in a Derivative operator. This violates the one-rule-per-step constraint and leaves the expression in an inconsistent state where one term is differentiated and the other is not.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 applies the chain rule to differentiate cos(4*x) and exp(4*x) but fails to apply the product rule to the first term (-sin(4*x)*exp(4*x)), which is still wrapped in a Derivative operator. This violates the one-rule-per-step constraint and leaves the expression in an inconsistent state where one term is differentiated and the other is not.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 incorrectly applies the chain rule to the first term: it replaces the derivative of −sin(4x) with the derivative of cos(4x). The correct derivative of −sin(4x) is −cos(4x)×4, not −sin(4x)×4. This single step violates the rule‑granularity requirement and leads to an incorrect final result.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 is labeled 'chain' but performs multiple differentiations (of cos(4x) and exp(4x)) simultaneously, violating the one-rule-per-step constraint. Additionally, Step 6 applies the product rule to the first term but fails to apply the chain rule to the resulting derivatives of sin(4x) and exp(4x), leaving them as undifferentiated Derivative(...) objects or implicitly skipping the chain rule application which is required for correctness.gpt-oss:20b: fail (error) 2026-09-20 — Step 6 applies both the product rule and the chain rule in one line, violating the rule that each step must change only one thing. The label "product" does not account for the chain rule used on the derivative of -sin(4*x).qwen3.6:27b-mlx: inconclusive 2026-09-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 5 is labeled 'chain' but performs the differentiation of cos(4x) and exp(4x) (which requires the 'derivative' rule for the outer functions combdeepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 6 applies the product rule incorrectly to the first term: the derivative of −4*cos(4x) is 16*sin(4x), not −sin(4x)*4. The step also fails to differentiate the second term, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: inconclusive 2026-09-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 5 is labeled 'chain' but performs the differentiation of cos(4x) and exp(4x) (which requires the 'derivative' rule for known forms combindeepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 5 applies two chain-rule substitutions in a single step, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 5 is labeled 'chain' but performs the differentiation of cos(4x) and exp(4x) using the chain rule, which is a differentiation step, not just a structural rule application like product or sum. More critically, the label 'chain' is not in the allowed vocabulary for differentiation steps (which should be 'derivative' for unfolding d/dx on known forms, or 'chain' is not listed, wait, 'chain' IS in the list: algebra, chain, constant, constant-multiple, derivative, exponential, inverse-trig, logarithmic, power, product, quotient, rewrite, simplify, sum). Let's re-read the vocabulary: 'chain' is in the list. Let's re-evaluate Step 5. It changes `Derivative(cos(4*x), x)` to `-4*sin(4*x)` and `Derivative(exp(4*x), x)` to `4*exp(4*x)`. This is applying the chain rule to compute derivatives. The label 'chain' is appropriate. Let's look at Step 4. It applies the product rule to the second term `Derivative((cos(4*x))*exp(4*x), x)`. It expands it to `Derivative(cos(4*x), x)*exp(4*x) + cos(4*x)*Derivative(exp(4*x), x)`. This is a correct application of the product rule. The label is 'product'. This is fine. Let's look at Step 6. It applies the product rule to the first term `Derivative((-sin(4*x))*exp(4*x), x)`. It expands it to `Derivative(-sin(4*x), x)*exp(4*x) + (-sin(4*x))*Derivative(exp(4*x), x)`. Wait, the output in step 6 is `((-4*cos(4*x))*exp(4*x) - sin(4*x)*(4*exp(4*x)))/8`. This step has ALREADY computed the derivatives of `-sin(4*x)` and `exp(4*x)`. It did not just apply the product rule structure; it also evaluated the derivatives. Step 4 only applied the product rule structure, leaving `Derivative(...)` terms. Step 5 evaluated the `Derivative` terms in the second part of the sum. Step 6 applies the product rule structure to the first part AND evaluates the derivatives inside it. Step 6 changes `Derivative((-sin(4*x))*exp(4*x), x)` to `(-4*cos(4*x))*exp(4*x) - sin(4*x)*(4*exp(4*x))`. This involves: 1. Product rule: `D(uv) = u'v + uv'`. 2. Chain rule/Derivative evaluation: `D(-sin(4x)) = -4cos(4x)` and `D(exp(4x)) = 4exp(4x)`. The label is 'product'. This is a defect because it applies more than one rule (product rule AND derivative evaluation/chain rule) in a single step. The contract says 'Each step changes ONE thing'. Step 6 changes the structure (product rule) AND evaluates the inner derivatives. Alternatively, one could argue Step 5 was 'chain' (evaluating derivatives) and Step 6 should have been 'product' (expanding the structure) followed by another 'chain' or 'derivative' step. But Step 6 does both. Also, Step 5 label 'chain' is acceptable. Step 6 label 'product' is insufficient because it also computes the derivatives. Therefore, Step 6 is a defect.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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.