Home›Calculus 1›Derivatives of exponential functions›Problem 2.453

Derivative of \( \displaystyle - e^{x - 1} \)

Problem 2.453 · easy Mental math Video

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 \( - e^{x - 1} \)

Top: the graph of f = -e^(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -e^(x - 1).

✓ 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. How we verify

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.

Senior review claude-sonnet-5-5, 2026-10-08: pass — The derivative -exp(x-1) is correct. The constant-multiple step is labelled 'constant', which is acceptable, and the exponential step legitimately includes the trivial inner derivative. SymPy proved every line.

gpt-oss:20b: dismiss — Line 1 only restates the problem and is not a transformation, so it needs no rule label; SymPy proved it matches the problem as stated.
qwen3.6:27b-mlx: dismiss — Evaluating the derivative of a simple inner function (x-1) is part of the exponential/chain step under the contract, so the 'exponential' label is legitimate and the result is correct.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-08 — The derivative -exp(x-1) is correct. The constant-multiple step is labelled 'constant', which is acceptable, and the exponential step legitimately includes the trivial inner derivative. SymPy proved every line.
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 2026-09-20
qwen3.6:27b-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.