Derivative of \( \displaystyle e^{\cos{\left(2 x - 1 \right)}} \)
Problem 2.336 · hard
Differentiate \( \displaystyle f(x) = e^{\cos{\left(2 x - 1 \right)}} \).
- \[ \frac{d}{d x} e^{\cos{\left(2 x - 1 \right)}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = e^{\cos{\left(2 x - 1 \right)}} \frac{d}{d x} \cos{\left(2 x - 1 \right)} \]chainApply the chain rule to the exponential function.✓ Proved
- \[ = - e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right) \]chainApply the chain rule to the cosine function.✓ Proved
- \[ = - e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \frac{d}{d x} 2 x \]algebraDistribute the derivative to the term 2*x.✓ Proved
- \[ = - 2 e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \]derivative simplifyDifferentiate the linear term 2*x. Rearrange the terms into a final simplified form.✓ Proved
Answer \( - 2 e^{\cos{\left(2 x - 1 \right)}} \sin{\left(2 x - 1 \right)} \)
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 |
| 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 incorrectly replaces Derivative(2*x - 1, x) with Derivative(2*x, x), omitting the derivative of the constant term -1. This applies more than one rule at once (simplifying a sum inside a derivative) and mislabels the operation. The correct step would directly evaluate Derivative(2*x - 1, x) to 2.deepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (13)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 4 incorrectly replaces Derivative(2*x - 1, x) with Derivative(2*x, x), omitting the derivative of the constant term -1. This applies more than one rule at once (simplifying a sum inside a derivative) and mislabels the operation. The correct step would directly evaluate Derivative(2*x - 1, x) to 2.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 4 is labeled 'algebra' but performs differentiation (distributing the derivative operator over a sum), which should be labeled 'sum' or 'derivative'. Step 4 also implicitly handles the constant term -1, but the label 'algebra' is incorrect for a differentiation step.gpt-oss:20b: fail (error) 2026-09-20 — Step 4 incorrectly replaces Derivative(2*x - 1, x) with Derivative(2*x, x), dropping the derivative of the constant term -1. The correct application would be Derivative(2*x - 1, x)=Derivative(2*x, x)-Derivative(1, x)=2-0=2. This step applies more than one rule at once and misstates the operation.qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 4 is labeled 'algebra' but performs a differentiation operation (distributing the derivative operator over a sum), which should be labeled 'sum' or 'derivative'. Step 5 is labeled 'derivative' but computes the derivative of a constant term (-1) implicitly or just evaluates the remaining term; strictly, the transition from step 3 to 5 involves applying the sum rule and then the power/constant rules, so splitting it into two steps with these labels is imprecise.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 4 incorrectly drops the derivative of the constant term –1. The derivative of 2*x-1 is 2, but the step claims to distribute the derivative only to 2*x, ignoring the constant. This violates the rule that each step must apply only one valid transformation.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 4 is labeled 'algebra' but performs a differentiation operation (distributing the derivative operator over a sum), which is not an algebraic simplification. The derivative of a sum should be handled by the 'sum' rule or combined into the previous chain rule step.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (error) 2026-09-18 — Step 4 incorrectly replaces Derivative(2*x - 1, x) with Derivative(2*x, x), dropping the constant term. This applies two rules at once (algebraic simplification and derivative of a constant) and changes the expression incorrectly. The correct derivative of 2*x - 1 is 2, so the step should have been Derivative(2*x, x) - Derivative(1, x) = 2 - 0 = 2.
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.