Home›Calculus 1›Chain rule›Problem 2.142

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

Problem 2.142 · medium

Differentiate \( \displaystyle f(x) = e^{x + 1} \).

\[ \frac{d}{d x} e^{x + 1} \]
Start with the derivative of the function.✓ Proved
\[ = e^{x + 1} \frac{d}{d x} \left(x + 1\right) \]
chainApply the chain rule.✓ Proved
\[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} x\right) e^{x + 1} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = e^{x + 1} \]
derivative simplifyThe derivative of x is 1 and the derivative of 1 is 0. Simplify the final expression.✓ Proved

Answer \( e^{x + 1} \)

✓ 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.

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
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: fail (style) — First step lacks a rule label, which is required by the contract.
qwen3.6:27b-mlx: pass

Every verdict on record (15)

qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (style) 2026-09-20 — Step 1 lacks an explicit rule label; it should be labeled "derivative" to match the allowed vocabulary. All other steps correctly apply a single rule each.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: fail (style) 2026-09-19 — First step lacks a rule label, which is required by the contract.
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (style) 2026-09-19 — Step 1 lacks a rule label from the allowed vocabulary; it should be labeled "derivative".
qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies the chain rule, sum rule, and basic derivative rules in separate steps. The labels are accurate and the logic is sound.
deepseek-r1:70b: fail (style) 2026-09-18 — Step 1 is missing a rule label, which is required by the contract.
gpt-oss:20b: pass 2026-09-18
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.