Derivative of \( \displaystyle \frac{\left(- \sin{\left(2 x - 3 \right)} + \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \)
Problem 2.18 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(2 x - 3 \right)} + \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \).
- \[ \frac{d}{d x} \frac{\left(- \sin{\left(2 x - 3 \right)} + \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \]constantStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- e^{2 x - 3} \sin{\left(2 x - 3 \right)} + e^{2 x - 3} \cos{\left(2 x - 3 \right)}\right)}{4} \]sumDistribute the division by 4 and apply the sum rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- e^{2 x - 3} \sin{\left(2 x - 3 \right)}\right)}{4} + \frac{\frac{d}{d x} e^{2 x - 3} \cos{\left(2 x - 3 \right)}}{4} \]productApply the product rule to each term in the numerator.✓ Proved
- \[ = - e^{2 x - 3} \sin{\left(2 x - 3 \right)} \]chain algebra simplify simplifyApply the chain rule to the trigonometric and exponential functions. Remove unnecessary parentheses. Combine like terms in the numerator. Simplify the final expression.✓ Proved
Answer \( - e^{2 x - 3} \sin{\left(2 x - 3 \right)} \)
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 |
| 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 both the product rule and the chain rule at once, yet labels it only as "chain". Each step must change only one thing and name the rule correctly.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 2 applies the distributive property (algebra) and the sum rule (derivative) simultaneously, violating the one-rule-per-step constraint. Step 3 is labeled 'product' but performs the linearity of the derivative (sum rule) on the outer structure, while the actual product rule application occurs in Step 4.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 applies the distributive property (algebra) and the sum rule (derivative) simultaneously, violating the one-rule-per-step constraint. Step 3 is labeled 'product' but performs the linearity of the derivative (sum rule) on the outer structure, while the actual product rule application occurs in Step 4.gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies both the product rule and the chain rule at once, yet labels it only as "chain". Each step must change only one thing and name the rule correctly.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 claims to apply the 'sum' rule, but it actually distributes the constant factor 1/4 into the terms (algebra/constant-multiple) and splits the derivative of a sum into a sum of derivatives (sum rule) simultaneously. Step 3 claims to apply the 'product' rule, but it merely distributes the derivative operator over the sum established in step 2 (sum rule); the actual product rule application happens in step 4. The labels in steps 2 and 3 do not match the primary operation performed in those specific transitions.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 applies two rules at once: it distributes the constant factor 1/4 (constant-multiple) and splits the sum (sum), violating the one-rule-per-step constraint. Step 3 is also defective because it applies the product rule to both terms simultaneously rather than handling them sequentially.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Steps 2 and 4 apply more than one rule at once (distribution/division and sum, product and chain). Each step should change only one rule, so the labeling is too coarse.qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 applies two rules at once (distributing the constant factor and splitting the sum), violating the one-rule-per-step constraint. Step 3 is labeled 'product' but performs no product rule expansion; it merely distributes the derivative operator, which should be labeled 'sum' or 'derivative'.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 2 mislabels the application of the product rule as the sum rule, which could mislead students about the appropriate rule to apply.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 2 applies both the constant multiple rule (distributing the 1/4) and the sum rule (splitting the derivative) simultaneously, violating the one-rule-per-step constraint. Step 3 is labeled 'product' but does not apply the product rule; it merely distributes the derivative operator over the sum, which is the sum rule.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (error) 2026-09-18 — Step 4 applies both the product rule and the chain rule in a single step, yet it is labeled only as "chain". Each step must change only one thing and use the correct rule name. This violates the contract.gpt-oss:20b: fail 2026-09-17 — Step 4 is labeled as applying only the chain rule, but it actually applies both the product rule (to each product) and the chain rule (to the inner sin, cos, and exp). This mislabeling could mislead a student into thinking only the chain rule is needed.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.