confused about how to get the fourth function

[valid vortex]

i got the answer but i have no idea where the first -1 comes from
it is -1(-1)^n(n!)/(x^(n+1))

i thought it wouldve just been (-1)^n(n!)/(x^(n+1))

?

[blazing spire]

(-1)^n*n!/x^n

[coarse lava]

ah, this is just taylor series

the first -1 comes in when you plug in 1.
-1(-1)^1 = 1

which is what you get as f'(x)

[blazing spire]

My answer is right?

[sudden charm]

f(x) = negative 1/x is where the initial - comes from

had f(x) been 1/x instead, (-1)^n(n!)/(x^(n+1)) wouldve been correct