Suppose all the derivatives of some function f exist at 0 and that the Taylor series for f about x = 0 is given by: x + x^2/2 + x^3/3 + x^4/4 + . . .

Give the value of each of the following.
f'(0):
f"(0):
f"'(0):
f^10(0):

Question

Suppose all the derivatives of some function f exist at 0 and that the Taylor series for f about x = 0 is given by: x + x^2/2 + x^3/3 + x^4/4 + . . .

Give the value of each of the following.
f'(0):
f"(0):
f"'(0):
f^10(0):

anonymous · Answer

use the derivative each time and then set x = 0 to get the answer but gor f^10(0) i have no idea how to do it

anonymous · Answer

f'(0):1
f"(0):1
f"'(0):2

anonymous · Answer

for f^10(0) i got zero but it was wrong. what's the trick here

rogue · Answer

Notice that as you keep taking the value of the derivative at x = 0, you'll get 1, 1, 2, 6, etc. These are 0!, 1!, 2!, 3!,...(n-1)!, where n is the nth derivative you're taking. So the 10th derivative would be 9! = 362880.

Hope that helps =)