Derivative of \( \displaystyle e^{8 x} \)
Problem 2.374 · medium Mental math
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 \( 8 e^{8 x} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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.
Every verdict on record (6)
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.