Derivative of \( \displaystyle \frac{5 e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{5 e^{4 x} \cos{\left(4 x \right)}}{8} \)
Problem 2.1829 · hard
Differentiate \( \displaystyle f(x) = \frac{5 e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{5 e^{4 x} \cos{\left(4 x \right)}}{8} \).
- \[ \frac{d}{d x} \left(\frac{5 e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{5 e^{4 x} \cos{\left(4 x \right)}}{8}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{5 \frac{d}{d x} e^{4 x} \cos{\left(4 x \right)}}{8} \]constant-multipleFactor out the constant 5/8.✓ Proved
- \[ = - \frac{5 e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{8} - \frac{5 \cos{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} + \frac{5 \frac{d}{d x} e^{4 x} \sin{\left(4 x \right)}}{8} \]sumApply the difference rule.✓ Proved
- \[ = \frac{5 e^{4 x} \frac{d}{d x} \sin{\left(4 x \right)}}{8} - \frac{5 e^{4 x} \frac{d}{d x} \cos{\left(4 x \right)}}{8} + \frac{5 \sin{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} - \frac{5 \cos{\left(4 x \right)} \frac{d}{d x} e^{4 x}}{8} \]productApply the product rule to both terms.✓ Proved
- \[ = 5 e^{4 x} \sin{\left(4 x \right)} \]chain algebra algebra simplify simplifyApply the chain rule to the exponential and trigonometric functions. Distribute the negative sign. Simplify the double negative. Combine like terms. Final simplification.✓ Proved
Answer \( 5 e^{4 x} \sin{\left(4 x \right)} \)
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 3 applies both the sum (difference) rule and the product rule in a single line, violating the one‑rule‑per‑step rule. The label "sum" is also misleading because the product rule is actually applied to the second term. This makes the step incorrect under the contract.qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'sum' but applies the product rule to the second term, violating the one-rule-per-step constraint. Step 4 is labeled 'product' but applies the product rule to the first term while leaving the second term (already expanded in step 3) unchanged, which is logically inconsistent with the previous step's state.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 3 is labeled 'sum' but applies the product rule to the second term, violating the one-rule-per-step constraint. Step 4 is labeled 'product' but applies the product rule to the first term while leaving the second term (already expanded in step 3) unchanged, which is logically inconsistent with the previous step's state.gpt-oss:20b: fail (error) 2026-10-08 — Step 3 applies both the sum (difference) rule and the product rule in a single line, violating the one‑rule‑per‑step rule. The label "sum" is also misleading because the product rule is actually applied to the second term. This makes the step incorrect under the contract.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 3 is labeled 'sum' but applies the product rule to the second term, violating the one-rule-per-step constraint. Step 4 is labeled 'product' but applies the product rule to the first term while leaving the second term unchanged, which is inconsistent with the label implying a general application and violates the granularity rule by skipping the application to the first term in the previous step.
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 2026-10-08 with SymPy 1.14.0.