Home›Calculus 1›Derivatives of exponential functions›Problem 2.307

Derivative of \( \displaystyle \left(5 x - 5\right) e^{x} \)

Problem 2.307 · medium Video

Differentiate \( \displaystyle f(x) = 5 \left(x - 1\right) e^{x} \).

\[ \frac{d}{d x} \left(5 x - 5\right) e^{x} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 5 \frac{d}{d x} \left(x - 1\right) e^{x} \]
constant-multiplePull out the constant factor 5.✓ Proved
\[ = 5 \left(x - 1\right) \frac{d}{d x} e^{x} + 5 e^{x} \frac{d}{d x} \left(x - 1\right) \]
productApply the product rule to the remaining terms.✓ Proved
\[ = 5 \left(x - 1\right) \frac{d}{d x} e^{x} + 5 e^{x} \]
derivative simplifyDifferentiate the first part of the product (x - 1). Simplify the derivative of (x - 1).✓ Proved
\[ = 5 \left(x - 1\right) e^{x} + 5 e^{x} \]
exponentialDifferentiate the exponential term.✓ Proved
\[ = 5 x e^{x} \]
algebra simplify simplifyDistribute the exponential term. Combine like terms. Final simplified form.✓ Proved

Answer \( 5 x e^{x} \)

Top: the graph of f = (5x - 5)·e^(x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (5x - 5)·e^(x) + 5e^(x).

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

Senior review claude-sonnet-5-5, 2026-10-07: pass — The derivative is correct: 5*(exp(x) + (x-1)*exp(x)) simplifies to 5*x*exp(x). Every step changes one thing and its rule label is legitimate, and SymPy proved each line. No specific panel objection was listed, so there is nothing to rule on.

Every verdict on record (13)

claude-sonnet-5-5: pass 2026-10-07 — The derivative is correct: 5*(exp(x) + (x-1)*exp(x)) simplifies to 5*x*exp(x). Every step changes one thing and its rule label is legitimate, and SymPy proved each line. No specific panel objection was listed, so there is nothing to rule on.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple, product, and derivative rules in a step-by-step manner. The algebraic simplifications are valid and clearly labeled.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple, product, and derivative rules in a logical sequence. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 4 applies the derivative to both terms of (x‑1) in a single step, combining two derivative operations. It should be split into two separate derivative steps or labeled as a sum rule. This violates the one‑rule‑per‑step rule.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-17
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.