Derivative of \( \displaystyle e^{6 x} \)
Problem 2.379 · medium Mental math Video
- \[ \frac{d}{d x} e^{6 x} \]Start with the derivative of the function.✓ Proved
- \[ = 6 e^{6 x} \]exponentialApply the rule for differentiating the exponential function e^{u(x)}.✓ Proved
✓ 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 (error) — Step 2 applies both the exponential rule and the chain rule at once, but labels it only as "exponential". Each step must change only one thing and use a single, correctly named rule.qwen3.6:27b-mlx: fail (error) — Step 2 applies both the chain rule and the exponential derivative rule simultaneously to transform d/dx(exp(6x)) to 6*exp(6x). The contract requires each step to change only one thing; this step should have been split into applying the chain rule first, then the exponential derivative.
Senior review claude-sonnet-5-5, 2026-10-08: pass — d/dx exp(6x) = 6*exp(6x) is correct, and the 'exponential' label fits the rule applied (e^u with u' = 6). All lines were proved by SymPy, so no weak lines remain.
gpt-oss:20b: dismiss — The contract says evaluating a derivative of a simple inner function (here 6x) as part of a chain-rule step is part of that step, and 'exponential' is a legitimate label for d/dx e^{u}.qwen3.6:27b-mlx: dismiss — Differentiating exp(6x) to 6*exp(6x) is one exponential/chain step with a simple inner function, so splitting it is not required, and SymPy proved the line equal.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-08 — d/dx exp(6x) = 6*exp(6x) is correct, and the 'exponential' label fits the rule applied (e^u with u' = 6). All lines were proved by SymPy, so no weak lines remain.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 applies both the chain rule and the exponential derivative rule simultaneously to transform d/dx(exp(6x)) to 6*exp(6x). The contract requires each step to change only one thing; this step should have been split into applying the chain rule first, then the exponential derivative.gpt-oss:20b: fail (error) 2026-09-21 — Step 2 applies both the exponential rule and the chain rule at once, but labels it only as "exponential". Each step must change only one thing and use a single, correctly named rule.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 applies both the chain rule (differentiating the inner function 6x) and the exponential rule simultaneously, violating the one-rule-per-step constraint. It should be split into a chain rule step and an exponential rule step.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 has no rule label, and step 2 incorrectly labels the chain rule as "exponential".qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule via the exponential rule label, and the single step transformation is valid.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.