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Derivative of \( \displaystyle e^{8 x} \)

Problem 2.374 · medium Mental math Video

Differentiate \( \displaystyle f(x) = e^{8 x} \).

\[ \frac{d}{d x} e^{8 x} \]
Start with the derivative of the function.✓ Proved
\[ = 8 e^{8 x} \]
exponentialApply the rule for differentiating an exponential function.✓ Proved

Answer \( 8 e^{8 x} \)

Top: the graph of f = e^(8x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 8e^(8x).

✓ 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
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: fail (style) — Step 1 is unlabeled; according to the contract each step must name a rule from the fixed vocabulary. It should be labeled "derivative".
qwen3.6:27b-mlx: fail (error) — Step 2 applies the chain rule to differentiate the composite function exp(8x), but is labeled 'exponential'. The 'exponential' rule alone only handles d/dx exp(x); handling the inner derivative of 8x requires the chain rule.

Senior review claude-sonnet-5-5, 2026-10-08: pass — d/dx exp(8x) = 8*exp(8x) is correct, and SymPy proved each line. The 'exponential' label is acceptable, and 'chain' would also have fit.

gpt-oss:20b: dismiss — Line 1 only restates the problem and applies no rule, so it needs no label; it is not a step that changes anything.
qwen3.6:27b-mlx: dismiss — Differentiating exp(8x) with the exponential rule is legitimate. Evaluating the derivative of a simple inner function (8x) is part of the step, and when two labels fit, either is acceptable.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-08 — d/dx exp(8x) = 8*exp(8x) is correct, and SymPy proved each line. The 'exponential' label is acceptable, and 'chain' would also have fit.
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 applies the chain rule to differentiate the composite function exp(8x), but is labeled 'exponential'. The 'exponential' rule alone only handles d/dx exp(x); handling the inner derivative of 8x requires the chain rule.
gpt-oss:20b: fail (style) 2026-09-21 — Step 1 is unlabeled; according to the contract each step must name a rule from the fixed vocabulary. It should be labeled "derivative".
qwen3.6:27b-mlx: fail (error) 2026-09-20 — The step from line 1 to line 2 applies both the chain rule (to handle the inner function 8x) and the exponential rule simultaneously. The contract requires each step to change only one thing; applying two rules at once is a defect.
gpt-oss:20b: fail (style) 2026-09-20 — Step 1 is unlabeled; it should be marked with the rule "derivative". The solution is mathematically correct but the labeling is incomplete.
qwen3.6:27b-mlx: fail (error) 2026-09-20 — The step from line 1 to line 2 applies both the chain rule (to handle the inner function 8x) and the exponential rule simultaneously. This violates the contract that each step must change only one thing. The correct sequence would require an intermediate step applying the chain rule to produce 8 * exp(8x) * d/dx(8x), followed by a step differentiating the inner term.
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.