Derivative of \( \displaystyle e^{\left(5 x + 2\right)^{2} + 1} \)
Problem 2.471 · hard
Differentiate \( \displaystyle f(x) = e^{\left(5 x + 2\right)^{2} + 1} \).
- \[ \frac{d}{d x} e^{\left(5 x + 2\right)^{2} + 1} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = e^{\left(5 x + 2\right)^{2} + 1} \frac{d}{d x} \left(\left(5 x + 2\right)^{2} + 1\right) \]chainApply the chain rule.✓ Proved
- \[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(5 x + 2\right)^{2}\right) e^{\left(5 x + 2\right)^{2} + 1} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = e^{\left(5 x + 2\right)^{2} + 1} \frac{d}{d x} \left(5 x + 2\right)^{2} \]constant algebraThe derivative of a constant is zero. Simplify the expression.✓ Proved
- \[ = \left(10 x + 4\right) e^{\left(5 x + 2\right)^{2} + 1} \frac{d}{d x} \left(5 x + 2\right) \]powerApply the power rule and chain rule.✓ Proved
- \[ = \left(50 x + 20\right) e^{\left(5 x + 2\right)^{2} + 1} \]derivative constant-multiple algebraDifferentiate the linear term. Multiply the constants. Distribute and simplify the final expression.✓ Proved
Answer \( \left(50 x + 20\right) e^{\left(5 x + 2\right)^{2} + 1} \)
✓ 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.
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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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: passqwen3.6:27b-mlx: fail (style) — Step 6 applies both the power rule and the chain rule simultaneously, violating the single-rule-per-step constraint. Step 8 uses 'constant-multiple' to describe arithmetic simplification of constants, which is better labeled as 'algebra' or 'simplify'.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 6 applies both the power rule and the chain rule simultaneously, violating the single-rule-per-step constraint. Step 8 uses 'constant-multiple' to describe arithmetic simplification of constants, which is better labeled as 'algebra' or 'simplify'.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a step applying the power rule to get 2*(5*x+2)*Derivative(5*x+2, x) is incorrect because the power rule alone yields 2*(5*x+2)^(1) * Derivative(5*x+2, x) only if the chain rule is also invoked; strictly, the power rule on u^2 is 2u*u', so step 6 is a compound step.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 8 uses 'constant-multiple' to describe arithmetic simplification (2*5=10), which is better labeled 'algebra' or 'simplify'.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.