Home›Calculus 1›Derivatives of exponential functions›Problem 2.25

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

Problem 2.25 · medium

Differentiate \( \displaystyle f(x) = \left(x - 1\right) e^{x} \).

\[ \frac{d}{d x} \left(x - 1\right) e^{x} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} x e^{x} - \frac{d}{d x} e^{x} \]
algebra rewrite simplifyDistribute the exponential term. Rewrite 1 as 1*exp(x) to prepare for the product rule. Simplify the expression.✓ Proved
\[ = x \frac{d}{d x} e^{x} + e^{x} \frac{d}{d x} x - \frac{d}{d x} e^{x} \]
productApply the product rule to the first term.✓ Proved
\[ = x \frac{d}{d x} e^{x} + e^{x} - \frac{d}{d x} e^{x} \]
derivativeDifferentiate x.✓ Proved
\[ = x e^{x} \]
derivative simplifyDifferentiate exp(x). Combine like terms.✓ Proved

Answer \( x e^{x} \)

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
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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 claims to use 'algebra' to transform the derivative of a product into a difference of derivatives, but this is an application of the linearity of the derivative operator (sum rule), not a purely algebraic manipulation of the integrand. Furthermore, Step 3 and Step 4 are redundant and logically circular, rewriting 1 as 1*exp(x) and then simplifying back, which adds no value and confuses the logical flow.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 claims to use 'algebra' to transform the derivative of a product into a difference of derivatives, which is a differentiation rule (linearity/sum rule), not an algebraic manipulation. Furthermore, Step 2 is mathematically incorrect as it drops the derivative of the first term's factor incorrectly or implies a non-existent rule.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 incorrectly applies the linearity of the derivative operator to the product (x-1)*exp(x) by splitting it into Derivative(x*exp(x)) - Derivative(exp(x)). The derivative of a product is not the product of derivatives, nor can the product be split this way before differentiation. The correct expansion would be Derivative(x*exp(x) - exp(x), x), followed by the sum rule.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 2 incorrectly rewrites (x-1)*exp(x) as x*exp(x)-exp(x) instead of x*exp(x)-1*exp(x). The missing factor of 1 causes an algebraic error that propagates through the rest of the solution.
gpt-oss:20b: fail 2026-09-17 — Step 3’s note incorrectly claims the rewrite prepares for the product rule, which is unnecessary; Step 6 incorrectly labels a simplification as a derivative and misstates the operation.
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.