# 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

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