Second derivatives
Problem 2.749 · medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{2} e^{- x} \).
- \[ \frac{d}{d x} x^{2} e^{- x} = x \left(2 - x\right) e^{- x} \]The first derivative.✓ Proved
- \[ \frac{d}{d x} x \left(2 - x\right) e^{- x} = \left(x^{2} - 4 x + 2\right) e^{- x} \]Differentiate again.✓ Proved
Answer \( \left(x^{2} - 4 x + 2\right) e^{- x} \)
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 explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | matches a second difference quotient of f at x = 0.7 |
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
generator:structured/second_derivative, checked 2026-09-26 with SymPy 1.14.0.