th derivative of $ frac {e ^ x – 1} {x} $ (with finite series)

Consider the function $ g (x) = frac {e ^ x – 1} {x} $. Find a general formula for $ g ^ {(n)} (x) $and prove that this formula is correct.

(Note: do not use Taylor series or infinite series)

I look at $ g ^ {(1)}, g ^ {(2)}, g ^ {(3)} $… and come with this fomular

$ frac {d ^ n} {dx ^ n} frac {e ^ x – 1} {x} $=$ frac {e ^ xn! (- 1) ^ n + n! (- 1) ^ {n + 1}} {x ^ {n + 1}} +
frac {e ^ x} {x ^ {n + 1}} sum_ {j = 1} ^ {n} frac {x ^ jn! (- 1) ^ {n-j}} {j!} $

I checked $ g ^ {(1)}, g ^ {(2)}, g ^ {(3)} $ for that they are correct.

But when I try to prove this induction of use, I find

$ frac {d} {dx} frac {e ^ xk! (- 1) ^ k + k! (- 1) ^ {k + 1}} {x ^ {k + 1}} neq frac {e ^ x (k + 1)! (- 1) ^ {k + 1} + (k + 1)! (- 1) ^ {(k + 1) +1}} {x ^ {(k + 1) +1}} $