Derivative of \( \displaystyle - e^{x - 1} \)
Problem 2.453 · easy Mental math
Differentiate \( \displaystyle f(x) = - e^{x - 1} \).
- \[ \frac{d}{d x} \left(- e^{x - 1}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} e^{x - 1} \]constantPull the constant factor out of the derivative.✓ Proved
- \[ = - e^{x - 1} \]exponentialApply the rule for differentiating the exponential function.✓ Proved
Answer \( - e^{x - 1} \)
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 |
| 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 (style) — Step 1 lacks a rule label; it should be marked "derivative". This omission violates the requirement that each step explicitly names the rule it applies.qwen3.6:27b-mlx: fail (error) — Step 3 claims to apply the 'exponential' rule but fails to apply the chain rule for the inner function (x-1). While the numerical result is coincidentally correct because the derivative of (x-1) is 1, the step is logically incomplete and misleading as it omits a required rule application.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 claims to apply the 'exponential' rule but fails to apply the chain rule for the inner function (x-1). While the numerical result is coincidentally correct because the derivative of (x-1) is 1, the step is logically incomplete and misleading as it omits a required rule application.gpt-oss:20b: fail (style) 2026-09-21 — Step 1 lacks a rule label; it should be marked "derivative". This omission violates the requirement that each step explicitly names the rule it applies.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is mathematically incorrect because it omits the application of the chain rule to the inner function (x - 1). The derivative of exp(x - 1) is exp(x - 1) * 1, but the step jumps directly to the result without accounting for the inner derivative, effectively treating the inner function as if its derivative were 1 without showing the chain rule step.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies the chain rule to differentiate exp(x-1) but is labeled 'exponential', which only covers the derivative of exp(x). The inner derivative of (x-1) is missing from the logic, making the step mathematically incomplete despite the algebraic coincidence.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 applies the derivative rule but is labeled "unlabelled". Each step should name the rule it applies; the missing label is a stylistic defect.
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.