Home›Calculus 1›Derivatives of exponential functions›Problem 2.492

Derivative of \( \displaystyle e^{9 x^{2} + 1} \)

Problem 2.492 · hard Video

Differentiate \( \displaystyle f(x) = e^{9 x^{2} + 1} \).

\[ \frac{d}{d x} e^{9 x^{2} + 1} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = e^{9 x^{2} + 1} \frac{d}{d x} \left(9 x^{2} + 1\right) \]
chainApply the chain rule.✓ Proved
\[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} 9 x^{2}\right) e^{9 x^{2} + 1} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = 9 e^{9 x^{2} + 1} \frac{d}{d x} x^{2} \]
derivativeDifferentiate each term separately.✓ Proved
\[ = 18 x e^{9 x^{2} + 1} \]
power simplifyApply the power rule to x**2. Simplify the final expression.✓ Proved

Answer \( 18 x e^{9 x^{2} + 1} \)

Top: the graph of f = e^(9x² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 18x·e^(9x² + 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
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
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
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-09: fail — The math is correct and every line is proved. Step 4 does two independent things, pulling the 9 out of the first term (constant-multiple) and replacing d(1)/dx with 0 (constant), under the generic label 'derivative'. It should be split into two steps. This is a style issue only.

Every verdict on record (7)

claude-sonnet-5-5: fail (style) 2026-10-09 — The math is correct and every line is proved. Step 4 does two independent things, pulling the 9 out of the first term (constant-multiple) and replacing d(1)/dx with 0 (constant), under the generic label 'derivative'. It should be split into two steps. This is a style issue only.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies both the constant‑multiple rule and the derivative rule at once, violating the one‑rule‑per‑step requirement. The correct labeling would be "constant‑multiple" for extracting the 9 and then a separate "derivative" step for x**2.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20

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.