Derivative of \( \displaystyle \frac{3 e^{2 x + 1}}{2} \)
Problem 2.546 · medium Video
Differentiate \( \displaystyle f(x) = \frac{3 e^{2 x + 1}}{2} \).
- \[ \frac{d}{d x} \frac{3 e^{2 x + 1}}{2} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} e^{2 x + 1}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{3 e^{2 x + 1} \frac{d}{d x} \left(2 x + 1\right)}{2} \]chainApply the chain rule.✓ Proved
- \[ = 3 e^{2 x + 1} \]derivative algebraDifferentiate the inner function. Simplify the expression by multiplying the constants.✓ Proved
Answer \( 3 e^{2 x + 1} \)
✓ 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 |
| 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; it should be marked with the rule "derivative". No other rule violations are present.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (style) 2026-09-26 — Step 1 is unlabeled; it should be marked with the rule "derivative". No other rule violations are present.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The labels are accurate and the algebraic simplification is correct.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 1 is unlabeled; it should be marked with the rule "derivative". All other steps correctly apply a single rule and are properly labeled.
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.