Derivative of \( \displaystyle - \frac{\sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{5} \)
Problem 2.118 · hard
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{5} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{5 x + 2} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)} + \cos{\left(5 x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} e^{5 x + 2}\right)}{5} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} \left(5 x + 2\right) + e^{5 x + 2} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}\right)}{5} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- e^{5 x + 2} \sin{\left(5 x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} \left(5 x + \frac{\pi}{4} + 2\right) + 5 e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}\right)}{5} \]exponentialDifferentiate the exponential function.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- 5 e^{5 x + 2} \sin{\left(5 x + \frac{\pi}{4} + 2 \right)} + 5 e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}\right)}{5} \]chain derivativeApply the chain rule to the cosine term. Evaluate the derivatives of the inner linear functions.✓ Proved
- \[ = \sqrt{2} e^{5 x + 2} \sin{\left(5 x + \frac{\pi}{4} + 2 \right)} - \sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)} \]algebraDistribute the constant factor and simplify.✓ Proved
- \[ = \sqrt{2} \left(\sin{\left(5 x + \frac{\pi}{4} + 2 \right)} - \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}\right) e^{5 x + 2} \]simplifyFactor out the common terms.✓ Proved
Answer \( 2 e^{5 x + 2} \sin{\left(5 x + 2 \right)} \)
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: fail (error) — Step 5 applies both the exponential rule (d/dx exp(u)=exp(u)u') and the chain rule for the cosine term in a single line, violating the one‑rule‑per‑step rule.deepseek-r1:70b: fail (error) — The solution does not reach the stated answer and lacks necessary simplification steps.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final simplification to the stated answer is mathematically valid via trigonometric identities, though the transition from step 9 to the final answer is implicit, the provided steps are sound.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final simplification to the stated answer is mathematically valid via trigonometric identities, though the transition from step 9 to the final answer is implicit, the provided steps are sound.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies both the exponential rule (d/dx exp(u)=exp(u)u') and the chain rule for the cosine term in a single line, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 is labeled 'exponential' but performs two distinct operations: it evaluates the derivative of the inner function (5*x+2) to 5 (which is a 'derivative' step) and differentiates the outer exponential function (which is 'exponential'). The contract requires each step to change only one thing. Furthermore, Step 7 is labeled 'derivative' but actually performs algebraic simplification (multiplying by 5 and distributing negatives), which should be labeled 'algebra' or 'simplify'. The label 'derivative' implies unfolding d/dx on a known form, which is not what happens in step 7.gpt-oss:20b: fail (error) 2026-09-20 — The final expression sqrt(2)*exp(5*x+2)*(sin(5*x+pi/4+2)-cos(5*x+pi/4+2)) is not algebraically equivalent to the stated answer 2*exp(5*x+2)*sin(5*x+2). The solution therefore contains a mathematical error.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final simplification correctly reduces the expression to the stated answer.deepseek-r1:70b: fail (error) 2026-09-19 — The solution does not reach the stated answer and lacks necessary simplification steps.gpt-oss:20b: fail (error) 2026-09-19 — The final expression does not match the correct derivative; the simplification step omitted the constant π/4+2 inside the trigonometric functions and the resulting coefficient is incorrect.qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 is labeled 'exponential' but it performs two distinct operations: it evaluates the derivative of the inner linear function (5*x+2) and applies the chain rule to the cosine term. This violates the one-rule-per-step constraint. Additionally, Step 7 is labeled 'derivative' but performs algebraic simplification (multiplying by 5), which should be labeled 'algebra' or 'simplify'.deepseek-r1:70b: fail (error) 2026-09-19 — The first step lacks a rule label, and the final answer does not align with the intermediate steps, indicating a possible error in simplification.gpt-oss:20b: fail (error) 2026-09-19 — The solution applies multiple rules in a single step (e.g., step 6 combines a chain rule with a derivative of a linear function, and step 7 repeats a derivative that was already performed). Moreover, the final expression sqrt(2)*exp(5*x+2)*(sin(5*x+pi/4+2)-cos(5*x+pi/4+2)) is not algebraically equivalent to the stated answer 2*exp(5*x+2)*sin(5*x+2).qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 5 is labeled 'exponential' but applies the chain rule to the exponential term (introducing Derivative(5*x+2, x)), which was not present in Step 4. Step 4 was labeled 'chain' but did not actually apply the chain rule (it left the derivative of the exponential as an unevaluated operator). The steps swap the application of the chain rule and the evaluation of the derivative, resulting in a logical inconsistency where the chain rule is claimed but not performed, or performed but not claimed.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (error) 2026-09-18 — Step 5 applies two rules at once: it differentiates the exponential term and the cosine term in a single line, violating the one‑change‑per‑step rule. The label "exponential" also does not cover the chain rule applied to the cosine.gpt-oss:20b: fail 2026-09-17 — The final answer 2*exp(5*x + 2)*sin(5*x + 2) does not match the derivative obtained in the steps, which is sqrt(2)*exp(5*x + 2)*(sin(5*x + pi/4 + 2) - cos(5*x + pi/4 + 2)).deepseek-r1:70b: fail 2026-09-17 — Steps 4 and 5 have misleading notes that do not fully describe the application of both the chain rule and differentiation to both exponential and cosine terms.
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.