Find the 600th derivative of f(x)=xe^(−x). Answer: f(600)(x)=?

f(x) = x e⁻ˣ

f'(x) = e⁻ˣ - x e⁻ˣ
f"(x) = -e⁻ˣ - (e⁻ˣ - x e⁻ˣ) = -2 e⁻ˣ + x e⁻ˣ
f'"(x) = 2 e⁻ˣ + (e⁻ˣ - x e⁻ˣ) = 3 e⁻ˣ - x e⁻ˣ
f⁽⁴⁾(x) = -3 e⁻ˣ - (e⁻ˣ - x e⁻ˣ) = -4 e⁻ˣ + x e⁻ˣ
------ (and so on)

It is found that the rule is: f⁽ⁿ⁾(x) = (-1)ⁿ⁺¹ n e⁻ˣ + (-1)ⁿ x e⁻ˣ
Hence, f ⁽⁶⁰⁰⁾(x) = (-1)⁶⁰¹ 600 e⁻ˣ + (-1)⁶⁰⁰ x e⁻ˣ

f ⁽⁶⁰⁰⁾(x) = -600 e⁻ˣ + x e⁻ˣ

Derivative of product

f(x) = x e^(-x)
f '(x) = e^(-x) - xe^(-x) = e^(-x) - f(x)
Differentiate again
f ''(x) = -e^(-x) - f'(x) = -2e^(-x) + xe^(-x) = -2e^(-x) + f(x)
And again
f ' ' '(x) = 2e^(-x) + f'(x) = 2e^(-x) + e^(-x) - f(x) = 3e^(-x) - f(x)

The coefficient next to e^(-x) is n if odd, and -n if even.
f(x) is being subtracted if odd, and added if even

Is 600 odd or even?
-600e^(-x) + f(x)
-600e^(-x) + x e^(-x)