Derivative of \( \displaystyle \frac{\left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \)
Problem 2.1447 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \).
- \[ \frac{d}{d x} \frac{\left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]Start with the derivative of the function.✓ Proved
- \[ = \left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right) \frac{d}{d x} \frac{e^{4 x - 3}}{8} + \frac{e^{4 x - 3} \frac{d}{d x} \left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right)}{8} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3} \frac{d}{d x} \left(4 x - 3\right)}{8} + \frac{e^{4 x - 3} \frac{d}{d x} \left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right)}{8} \]constant-multiplePull out the constant factor 1/8 from the second term's derivative.✓ Proved
- \[ = \frac{\left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{2} + \frac{e^{4 x - 3} \frac{d}{d x} \left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right)}{8} \]derivativeDifferentiate the inner function 4*x - 3.✓ Proved
- \[ = \frac{\left(- 4 \sin{\left(4 x - 3 \right)} - 4 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} + \frac{\left(- \sin{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{2} \]trig algebraDifferentiate the sine and cosine terms using the chain rule. Factor out -4 from the first term.✓ Proved
- \[ = \frac{\left(- 4 \sin{\left(4 x - 3 \right)} - 4 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} + \frac{\left(- 4 \sin{\left(4 x - 3 \right)} + 4 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]algebraSimplify the second term's coefficient.✓ Proved
- \[ = - e^{4 x - 3} \sin{\left(4 x - 3 \right)} \]algebra algebra simplify algebra simplifyCombine the terms under a common factor. Distribute the 4 into the parenthesis. Combine like terms inside the parenthesis. Multiply the terms. Cancel the 8 in the numerator and denominator.✓ Proved
Answer \( - e^{4 x - 3} \sin{\left(4 x - 3 \right)} \)
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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ 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: passqwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to differentiate the trigonometric terms but labels the step 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for differentiation rules. Since 'chain' is in the allowed vocabulary, the step must be labeled 'chain' (or 'derivative' if viewed as a composite derivative), making 'trig' an incorrect label for the operation performed.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 applies the chain rule to differentiate the trigonometric terms but labels the step 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for differentiation rules. Since 'chain' is in the allowed vocabulary, the step must be labeled 'chain' (or 'derivative' if viewed as a composite derivative), making 'trig' an incorrect label for the operation performed.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 applies the chain rule to differentiate the sine and cosine terms but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities, not differentiation rules. Additionally, the note claims the chain rule is used, but the label does not match the action.gpt-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.