Derivative of \( \displaystyle \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \)
Problem 2.28 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \).
- \[ \frac{d}{d x} \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]constant-multiplePull out the constant factor 1/8.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) \frac{d}{d x} e^{4 x - 3}}{8} + \frac{e^{4 x - 3} \frac{d}{d x} \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right)}{8} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) \frac{d}{d x} e^{4 x - 3}}{8} + \frac{\left(\frac{d}{d x} \sin{\left(4 x - 3 \right)} - \frac{d}{d x} \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]sumApply the sum rule to the first term.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) \frac{d}{d x} e^{4 x - 3}}{8} + \frac{\left(4 \sin{\left(4 x - 3 \right)} + 4 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]trig algebraDifferentiate the sine and cosine terms using the chain rule. Simplify the signs in the first term.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{2} + \frac{\left(4 \sin{\left(4 x - 3 \right)} + 4 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]chainDifferentiate the exponential term using the chain rule.✓ Proved
- \[ = e^{4 x - 3} \sin{\left(4 x - 3 \right)} \]algebra simplify simplifyFactor out the common term 4 * exp(4*x - 3). Combine like terms inside the parentheses. Simplify the coefficients and constants.✓ 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 |
| 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 7 incorrectly labels the derivative of the exponential as "chain"; the proper rule name from the vocabulary is "exponential". No mathematical errors occur, only a mislabeling of the applied rule.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to differentiate sin and cos, but labels the step 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not differentiation. Furthermore, the note claims to use the chain rule, which is a differentiation rule, but the label does not match the action (differentiation requires 'derivative' or 'chain' depending on granularity, but 'trig' is incorrect for differentiation). Specifically, the step performs differentiation, so 'trig' is a mislabel for a differentiation step.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 applies the chain rule to differentiate sin and cos, but labels the step 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not differentiation. Furthermore, the note claims to use the chain rule, which is a differentiation rule, but the label does not match the action (differentiation requires 'derivative' or 'chain' depending on granularity, but 'trig' is incorrect for differentiation). Specifically, the step performs differentiation, so 'trig' is a mislabel for a differentiation step.gpt-oss:20b: fail (style) 2026-09-20 — Step 7 incorrectly labels the derivative of the exponential as "chain"; the proper rule name from the vocabulary is "exponential". No mathematical errors occur, only a mislabeling of the applied rule.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications in a logical sequence. Each step adheres to the single-rule constraint and uses valid labels from the provided vocabulary.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is correct.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 5 labels the rule as "trig" but the operation actually applies the chain rule to differentiate the sine and cosine terms; the rule label does not match the action performed.deepseek-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.