Question Detail

First, find the first derivative f'(x) using the product rule (uv)' = u'v + uv'.Let u = x and v = e⁻ˣ.Then u' = 1 and v' = -e⁻ˣ.f'(x) = (1)(e⁻ˣ) + (x)(-e⁻ˣ) = e⁻ˣ - x e⁻ˣ = e⁻ˣ(1 - x).Now, find the second derivative f''(x) by differentiating f'(x) again using the product rule.Let u = e⁻ˣ and v = (1 - x).Then u' = -e⁻ˣ and v' = -1.f''(x) = (-e⁻ˣ)(1 - x) + (e⁻ˣ)(-1)= -e⁻ˣ + x e⁻ˣ - e⁻ˣ= x e⁻ˣ - 2 e⁻ˣ= e⁻ˣ(x - 2).