Question Detail
Question
If f(x) = x e-x, find f''(x).
Correct Answer
Option C is the correct answer.
Detailed Explanation
First, find the first derivative f'(x) using the product rule (uv)' = u'v + uv':Let u = x, v = e-x.Then u' = 1, v' = -e-x.f'(x) = (1)(e-x) + (x)(-e-x) = e-x - xe-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, v = (1 - x).Then u' = -e-x, v' = -1.f''(x) = (-e-x)(1 - x) + (e-x)(-1)= -e-x + xe-x - e-x= xe-x - 2e-x= e-x(x - 2).
Hint
Apply the product rule twice to find the second derivative. Be careful with the chain rule for e-x.