Derivative of \( \displaystyle - \frac{5 \sqrt{2} e^{3 x - 1} \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)}}{6} \)
Problem 2.1392 · hard
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{3 x - 1} \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)}}{6} \).
- \[ \frac{d}{d x} \left(- \frac{5 \sqrt{2} e^{3 x - 1} \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)}}{6}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{5 \sqrt{2} \frac{d}{d x} e^{3 x - 1} \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)}}{6} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(e^{3 x - 1} \frac{d}{d x} \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)} + \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{3 x - 1}\right)}{6} \]productApply the product rule.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(e^{3 x - 1} \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} \left(3 x - 1\right) + e^{3 x - 1} \cos{\left(- 3 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} \left(- 3 x + \frac{\pi}{4} + 1\right)\right)}{6} \]chainApply the chain rule to both terms.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(3 e^{3 x - 1} \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)} - 3 e^{3 x - 1} \cos{\left(- 3 x + \frac{\pi}{4} + 1 \right)}\right)}{6} \]derivative algebraDifferentiate the inner functions. Simplify the expression inside the parentheses.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(e^{3 x - 1} \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)} - e^{3 x - 1} \cos{\left(- 3 x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]algebra simplifyFactor out the common constant 3. Simplify the fraction 15/6 to 5/2.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(\sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)} - \cos{\left(- 3 x + \frac{\pi}{4} + 1 \right)}\right) e^{3 x - 1}}{2} \]simplifyFactor out the exponential term.✓ Proved
Answer \( 5 e^{3 x - 1} \sin{\left(3 x - 1 \right)} \)
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 |
| 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 4 applies the chain rule to both derivative terms in a single step, violating the rule that each step must change only one thing. The step should be split into two separate chain applications.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and simplifies the expression to the stated answer. Each step adheres to the one-rule-per-step constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and simplifies the expression to the stated answer. Each step adheres to the one-rule-per-step constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-03 — Step 4 applies the chain rule to both derivative terms in a single step, violating the rule that each step must change only one thing. The step should be split into two separate chain applications.qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.