Derivative of \( \displaystyle - \frac{e^{4 x}}{4} \)
Problem 2.229 · medium
Differentiate \( \displaystyle f(x) = - \frac{e^{4 x}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{e^{4 x}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} e^{4 x}}{4} \]constantPull out the constant factor.✓ Proved
- \[ = - \frac{e^{4 x} \frac{d}{d x} 4 x}{4} \]chainApply the chain rule to the exponential function.✓ Proved
- \[ = - e^{4 x} \]derivative algebraDifferentiate the inner function 4*x. Simplify the expression by multiplying the constants.✓ Proved
Answer \( - e^{4 x} \)
✓ 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.
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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (style) — Step 2 uses the label 'constant' to pull out a constant factor, but the correct label from the vocabulary is 'constant-multiple'. Step 5 uses 'algebra' for simplification, which is acceptable, but the label in Step 2 is incorrect.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 uses the label 'constant' to pull out a constant factor, but the correct label from the vocabulary is 'constant-multiple'. Step 5 uses 'algebra' for simplification, which is acceptable, but the label in Step 2 is incorrect.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 uses the label 'constant' to pull out a constant factor, but the correct label from the vocabulary is 'constant-multiple'. Step 4 uses the label 'derivative' for a simplification step (4*x -> 4), but 'derivative' is reserved for unfolding d/dx on known forms; this step is better labeled 'algebra' or 'simplify' as it just evaluates the derivative of a linear term.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 2 is labeled 'constant' but performs the action of pulling out a constant factor, which corresponds to the 'constant-multiple' rule in the vocabulary. The label 'constant' typically refers to the derivative of a constant being zero.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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.