Home›Calculus 1›Derivatives of exponential functions›Problem 2.333

Derivative of \( \displaystyle e^{4 x^{2} + 1} \)

Problem 2.333 · hard Video

Differentiate \( \displaystyle f(x) = e^{4 x^{2} + 1} \).

\[ \frac{d}{d x} e^{4 x^{2} + 1} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = e^{4 x^{2} + 1} \frac{d}{d x} \left(4 x^{2} + 1\right) \]
chainApply the chain rule.✓ Proved
\[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} 4 x^{2}\right) e^{4 x^{2} + 1} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = 8 x e^{4 x^{2} + 1} \]
power constant simplifyDifferentiate the power term. The derivative of the constant 1 is 0. Simplify the final expression.✓ Proved

Answer \( 8 x e^{4 x^{2} + 1} \)

Top: the graph of f = e^(4x² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 8x·e^(4x² + 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
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-07: fail — The panel objection is not specified, so there is nothing to rule on. The math is correct and SymPy proved every line. Two labeling defects remain. Line 4 (labeled power) also turns Derivative(1, x) into 0, so it does two independent things, one per term. Line 5 (labeled constant, note 'the derivative of 1 is 0') only drops the '+ 0', which is algebra or simplify, because the constant's derivative was already evaluated in line 4.

Every verdict on record (14)

claude-sonnet-5-5: fail (style) 2026-10-07 — The panel objection is not specified, so there is nothing to rule on. The math is correct and SymPy proved every line. Two labeling defects remain. Line 4 (labeled power) also turns Derivative(1, x) into 0, so it does two independent things, one per term. Line 5 (labeled constant, note 'the derivative of 1 is 0') only drops the '+ 0', which is algebra or simplify, because the constant's derivative was already evaluated in line 4.
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: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 4 applies two rules at once (power and constant‑multiple) but is labeled only as "power". Each step must change only one thing and name the rule correctly.
qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 4 is labeled 'power' but applies the power rule to 4*x**2 while leaving the derivative of 1 untouched; strictly, the power rule applies to the term 4*x**2, but the step also implicitly handles the constant term's derivative (or rather, the previous step split them, so this step should only address the power term, but the label 'power' is acceptable for 4*x**2 -> 8*x. However, Step 5 is labeled 'constant' but performs the simplification 8*x + 0 -> 8*x, which is algebraic simplification, not the application of the constant rule (which was already applied in step 3/4 logic). Actually, looking closer: Step 3 splits the derivative into a sum. Step 4 computes Derivative(4*x**2, x) as 8*x using 'power'. It leaves Derivative(1, x) as is? No, Step 4 shows (8*x + 0). This implies Step 4 computed BOTH derivatives: 4*x**2 -> 8*x (power) and 1 -> 0 (constant). Applying two rules (power and constant) in one step is a defect. Step 5 then simplifies 8*x + 0 to 8*x, labeled 'constant', which is incorrect labeling for simplification.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: fail (error) 2026-09-18 — Step 4 applies both the constant‑multiple rule (derivative of 4·x²) and the power rule in one go, violating the one‑rule‑per‑step rule. Step 5 then drops the zero term, which is a simplification, yet it is labeled as a constant rule. Both steps combine multiple transformations and mislabel the applied rule.

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.