Question

Find the 1000th derivative of f(x) = xeˣ:

A) f^(1000)(x) = eˣ
B) f^(1000)(x) = x
C) f^(1000)(x) = x¹000
D) f^(1000)(x) = 0

Answer 1

The 1000th derivative of f(x) = xeˣ is 1000eˣ + xeˣ.

Step-by-step explanation:

To find the 1000th derivative of f(x) = xeˣ, we can generalize the pattern of its derivatives. Starting with f'(x) = eˣ + xeˣ, we can see that each subsequent derivative adds an extra x to the front. So the 2nd derivative is f''(x) = 2eˣ + xeˣ, the 3rd derivative is f '''(x) = 3eˣ + xeˣ, and so on. Based on this pattern, the 1000th derivative will be f^(1000)(x) = 1000eˣ + xeˣ.