You have that $F(t)$ is a constant ($f(x)$) plus$$-f(t)-(x-t)f'(t)-\frac{(x-t)^2}2f''(t)-\cdots-\frac{(x-t)^n}{n!}f^{(n)}(t).$$Forget the constant; if you differentiate it, you get $0$. Then:
- if you differentiate $-f(t)$, you get $-f'(t)$;
- if you differentiate $-(x-t)f'(t)$, you get $f'(t)-(x-t)f''(t)$;
- if you differentiate $-\frac{(x-t)^2}{2}f''(t)$, you get $(x-t)f''(t)-\frac{(x-t)^2}2f^{(3)}(t)$;
- $\vdots$
- if you differentiate $-\frac{(x-t)^n}{n!}f^{(n)}(t)(t)$, you get $\frac{(x-t)^{n-1}}{(n-1)!}f^{(n)}(t)-\frac{(x-t)^n}{n!}f^{(n+1)}(t)$.
Now, sum up all of this. Everything gets cancelled, except for $-\frac{(x-t)^n}{n!}f^{(n+1)}(t)$.