If $f(x)=\frac{1}{1-x-x^2}$, find $\frac{f^{(10)}(0)}{10!}$ My math teacher posed this question to my calculus class. If $f(x)=\frac{1}{1-x-x^2}$, find $\frac{f^{(10)}(0)}{10!}$.
At first, I began by taking the first few derivatives, but it soon go out of hand with the repeated quotient rules. I'm sure that I could continue for $10$ derivatives, but I believe that there must be an easier solution.

Hint:
$$\frac1{1-u}=1+u+u^2+\dots+u^{10}+o(u^{10})$$
Set $u=xx+x^2$ and find the coefficient of $x^{10}$.
Another hint:
Perform the division of $1$ by $1-x-x^2$ along the increasing powers of $x$.
You should find the Fibonacci number $F_{11}$.
