Derivative of \( \displaystyle \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{4} \)
Problem 2.132 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{4} \).
- \[ \frac{d}{d x} \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{4} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{4} \]constant-multiplePull out the constant factor 1/4.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) \frac{d}{d x} e^{4 x - 3}}{4} + \frac{e^{4 x - 3} \frac{d}{d x} \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right)}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) \frac{d}{d x} e^{4 x - 3}}{4} + \frac{\left(\frac{d}{d x} \sin{\left(4 x - 3 \right)} - \frac{d}{d x} \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{4} \]sumApply the sum rule to the first term.✓ Proved
- \[ = \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3} + \frac{\left(4 \sin{\left(4 x - 3 \right)} + 4 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{4} \]trig algebraDifferentiate the sine and cosine terms using the chain rule. Simplify the signs.✓ Proved
- \[ = 2 e^{4 x - 3} \sin{\left(4 x - 3 \right)} \]algebra simplify simplifyFactor out 4 * exp(4*x - 3) from both terms. Cancel the 1/4 with the 4. Combine like terms inside the parentheses.✓ Proved
Answer \( 2 e^{4 x - 3} \sin{\left(4 x - 3 \right)} \)
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 |
| 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to differentiate the composite trigonometric and exponential functions, but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for the differentiation process which requires the 'chain' rule.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 applies the chain rule to differentiate the composite trigonometric and exponential functions, but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for the differentiation process which requires the 'chain' rule.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies both the trigonometric derivative rule and the chain rule, but labels it only as "trig". Each step must change only one rule, so this is a defect.qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 5 applies the chain rule to differentiate the composite trigonometric and exponential functions, but is labeled 'trig'. The label 'chain' is available in the vocabulary and more accurately describes the operation performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 5 incorrectly labels the application of the chain rule as "trig". The derivative of sin(4*x-3) and cos(4*x-3) requires the chain rule, not a trig rule. This labeling mismatch is a style defect.qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 applies the chain rule to differentiate sin(4x-3) and cos(4x-3), but labels the step 'trig'. The label 'trig' is reserved for trigonometric identities, not differentiation rules. Additionally, the step performs multiple differentiations at once, violating the one-change-per-step rule.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 5 is labeled "trig" but it applies the chain rule to differentiate sin and cos; the correct label should be "chain". No mathematical errors are present, only a mislabeling of the rule used.qwen3.6:27b-mlx: fail (misleading) 2026-09-18 — Step 5 is labeled 'trig', but it applies the chain rule to differentiate composite functions; the label should be 'chain' or 'derivative'. Additionally, the note explicitly mentions the chain rule, contradicting the label.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (style) 2026-09-18 — Step 5 applies the chain rule to differentiate sin(4*x‑3) and cos(4*x‑3), but it is labeled "trig". The correct label for this operation is "chain". No other rule is mis‑applied.gpt-oss:20b: fail 2026-09-17 — Step 5 applies both the chain rule and the derivative of a trigonometric function, but is labeled only as "trig" and the note does not mention the chain rule. This mislabeling and combining of two rules in one step could mislead a student.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.