Derivative of \( \displaystyle \frac{\left(5 \sin{\left(2 x + 1 \right)} - 5 \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \)
Problem 2.1134 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \).
- \[ \frac{d}{d x} \frac{\left(5 \sin{\left(2 x + 1 \right)} - 5 \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) \frac{d}{d x} e^{2 x + 1}}{4} + \frac{5 e^{2 x + 1} \frac{d}{d x} \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right)}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{5 \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{2} + \frac{5 \left(2 \sin{\left(2 x + 1 \right)} + 2 \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]chainApply the chain rule to the sine and cosine terms.✓ Proved
- \[ = \frac{5 \left(2 \sin{\left(2 x + 1 \right)} - 2 \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} + \frac{5 \left(2 \sin{\left(2 x + 1 \right)} + 2 \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \]algebraSimplify the derivative of the trigonometric part.✓ Proved
- \[ = 5 e^{2 x + 1} \sin{\left(2 x + 1 \right)} \]algebra algebra simplifyDistribute the exponential term. Combine like terms. Final simplification.✓ Proved
Answer \( 5 e^{2 x + 1} \sin{\left(2 x + 1 \right)} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 |
| 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: pass — The solution correctly applies the constant multiple, product, and chain rules in separate steps. The algebraic simplifications are valid and clearly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the constant multiple, product, and chain rules in separate steps. The algebraic simplifications are valid and clearly labeled.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 4 applies the chain rule to both the trigonometric terms and the exponential term, but the label 'chain' only explicitly references the trigonometric terms in the note, and the step combines multiple differentiation actions (derivative of sin, cos, and exp) into one step. While the algebra is correct, the step violates the 'one rule at a time' principle by differentiating three distinct composite functions simultaneously under a single label that doesn't fully capture the scope (specifically the exponential derivative). A stricter interpretation requires separating the differentiation of the exponential term or labeling it more broadly if allowed, but given the fixed vocabulary, applying 'chain' to the whole step is imprecise because it also involves the basic derivative of exp. However, the most significant defect is that Step 4 performs the differentiation of the exponential term `exp(2*x+1)` as well, which is not mentioned in the note, and effectively applies the chain rule to three different sub-expressions in one go. This violates the granularity constraint.gpt-oss:20b: fail (style) 2026-09-28 — Step 4 applies the chain rule to both the sine and cosine terms in a single step, violating the rule that each step must change only one thing. The label "chain" is appropriate, but the step combines two applications of the chain rule at once.
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 2026-09-28 with SymPy 1.14.0.